The goalkeeping question is one of a long list of defensive conundrums for Postecoglou to resolve. With Barkas at fault in his only on target engagement versus Midtjylland, followed by poor positioning in conceding to West Ham United, his fierce critics are even more vociferous than normal.
Goalkeeping data is similar to defending data. We miss the impact of e.g. correct positioning or good communication. That is, danger avoided through actions the event data cannot pick up.
Of course, over a large enough sample, other data can infer this. But as a word of caution, goalkeeping data is also a reflection of the strength of the defence in front. Last season that was suboptimal as was discussed at length, and this season so far it has comprised a very raw and inexperienced back line.
The other caution is simply sample size. Since collecting data for Celtic, there are only five goalkeepers who have played enough minutes to warrant inclusion in a bench marking exercise – Bain, Barkas, De Vries, Forster and Gordon.
Clearly, a small sample.
xG Data
The most important xG concept for goalkeepers is Post Shot xG. The xG you see reported on most sites is generally the xG based on the position and context the ball was struck from. Post Shot xG is taken at the point the ball reaches the goal, with context and position of where is started from taken into account. Simply, the Post Shot xG will be higher than the initial xG by the fact that all blocked and off target shots will be filtered out.
By comparing Post Shot xG with goals conceded, we can asses the extent to which a goalkeeper has saved the shots expected or not.
Note that for the first three seasons of Gordon’s Celtic career I do not have comprehensive xG data so he will suffer by comparison given his form in those seasons was better than in his last three seasons.

What this means is, per 90m, Bain has conceded 0.08 less goals than the Expected Goals model would predict. This is slightly ahead of Forster (essentially the 19/20 season).
Barkas is conceding 0.18 goals per 90m more than expected.
Forster is a fraction behind Bain and this illustrates perhaps that whilst Forster is a very good goalkeeper, he has not been first choice at Southampton for many years, and by performing very well in high profile games, it is perhaps that we remember over the more mundane matches.
Save %
Most outlets will quote Save % at you. I include this for completeness as it is another useful data point but not as instructive as the one above as it simply does not account for shot quality. It shows the % of on target shots saved.

Whilst it is a highly flawed metric, nevertheless the difference between Forster and Gordon on 74% and De Vries (56%) and Barkas (51%) is stark. Across all saves in this period from all ‘keepers (last 7 seasons), Bain is par with the average save rate. At the moment that feels appealing!
Save Volume
This is another metric to be careful about as it tells you as much about the defensive protection as individual performance. But you may be interested how few saves Celtic ‘keepers have to make:

Barkas is making around half the volume of saves as Gordon, Forster or Bain. Given the context of his appearances last season – behind a creaking defence that was poorly protected, that is not good.
Defensive Errors
Errors are subjective but as I am the sole arbiter of them, there is at least consistency to the subjectivity!
Barkas has attracted further criticism from some Celtic fans in this regard yet, to my assessment, has the lowest error rate of any Celtic ‘keeper at 0.05 per 90m. Indeed, his most glaring error until the Midtjylland game was the free kick goal AC Milan scored last season where his footwork was all wrong.
Some have blamed him for 2 goals last season against The Rangers – the Goldson header and the McGregor own goal. In my opinion he couldn’t be blamed for either but I would be interested in a goalkeeping expert such as @jhdharrison1 on Twitter for a second opinion. I feel this is classic displacement of disappointment onto the nearest and most obvious target.
In terms of differential between defensive saves (interventions that saved a goal or great chance) vs defensive errors, the stats say:

Surprisingly, Gordon made slightly more great saves per 90m than Forster but probably not so much in very important games. Forster had the best differential between saves and errors.
As can be seen neither Bain nor Barkas are big on big saves, and if anything, Bain is more error prone.
Possession
Recoveries
Recoveries are where the player recovers the ball after a tackle or lose pass / clearance and the ball is out of any team’s possession. Good recovery numbers are indicative of anticipation, speed, awareness, positioning.

By this metric, Barkas is just behind Gordon.
Defensive 3rd Wins / Losses
This is the number of times per 90m the player either loses possession in their own defensive 3rd (e.g. mis placed pass, clearance, being tackled) versus wins the ball back in their own defensive 3rd (e.g. an interception, claiming a cross).

Barkas has the best differential between possession wins and losses in the defensive 3rd.
Crosses
One reason is that he has the best cross catching stats:

Bear in mind under Postecoglou, Celtic will surrender space wide on occasion, plus the nature of much of Scottish football is to bang crosses into the box. These, again, are small numbers but Barkas has the edge on catching crosses.
Distribution
Volume of passing from the ‘keeper and accuracy can say as much about the style and approach of the team as the individual player However, we can probably agree Celtic generally have played a passing style over the years with nuances.
In terms of simple pass accuracy:

There is a larger range than you might imagine with Barkas completing 88% of all passes and Forster only 67%. Some of that is to do with style but both played under Lennon.
De Vries was brought in because of his passing, and he couldn’t even do that well!
Another consideration with the “bring Forster back” clamour is the overall profile of the goalkeeper Celtic need. Postecoglou clearly needs a stopper with good feet and distribution as he will sometimes be the third centre back to create overloads elsewhere. Remember, a Celtic ‘keeper will generally only make 1-3 saves per 90m but be expected to attempt nearer 30 passes. So, what is the higher priority skill? (I realise the true answer is not binary).
Pack Passes
Strangely, Barkas does not have the highest Pack Pass rate – the number of passes per 90m that are forward and take opponents out the game. Again, the volumes are low here:

This is where Bain has a slight edge and Barkas perhaps takes fewer risks.
Conclusion
The ideal Celtic goalkeeper should:
High concentration – redundant for long periods
Shot stopper – perform at the level appropriate to upper European football
Distribution – be comfortable with ball at feet and of breaking the initial press
Possession – be secure coming to collect a high volume of opposition crosses
Low Error Rate – low possession means any error exacerbated
It’s a tough profile to scout for.
At the moment Barkas has not shown himself reliable to make the saves needed and indeed has only made outstanding saves in two games – away at Ross County in a 5-0 win and at home to St Johnstone in a 4-0 win. However, his distribution and cross catching skills are very good. He profiles well in some criteria.
Overall, neither incumbent is as safe in keeping the ball out as Gordon and Forster have been over the recent years. Bain is probably the more secure option and is good enough with his feet but also the most error prone.
Celtic should assess Barkas’s confidence but Postecoglou strikes me as a manager that will show faith and try and build his players confidence.
However, if I was basing the selection purely on the performance stats I have available, I’d be going with Bain for now whilst scouring the market.