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Lets assume we solved an abstract game with CFRM that contains the following bets: 0.5*PS, PS, 2*PS, All-In. Whenever we face a bet that deviates from our abstraction we need to find a mapping, which is discussed in some papers. With soft translation, we know that we should map a 0.75 PS bet probabilistically to a 0.5PS bet (50%) or a PS bet (50%).
Now how do we handle the following issues: assume our strategy tells us to call. However, given the pot was 10 before the bet, he bet 7.5 and we called, we have a pot size of 25, while the next node assumes we either have a 20 or a 30bb pot. The different stack sizes should obviously change our strategy a lot (i.e. the bigger the pot, the less we can fold).
The problem gets even bigger when we have another 0.75PS bet and map it to the same betting abstraction as before - the difference between real pot size and abstract pot size grows strongly. I saw some approaches that changes the size of raises for example, that the pot in the following node is again correct, however, first I think it might lead to wrong results (ie., raising big so that we are commited but then folding to a reraise) and second it doesn't work at all for calls.
How do you solve these issues?

I think the right way is to make effective stack size a part of abstraction and solve a game severa times

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I'm not sure there is a way to 'solve' the problem, outside of creating bigger abstractions that handle the game you're playing. You can apply 'fixes,' translations, and/or generalizations but you're ultimately just taking your best guess at what would be the 'real' strategy.

I don't understand what you exactly mean. I'm training a model for a certain eff. stack size, but even if the effective stack size is exactly right (e.g. first hand, 25bb each), the difference comes not from different stack sizes but from different actions taken. Or do you suggest to search games with other different stack sizes, look for the node with the same history and find the node with the best matching SPR?

And this is exactly why fixed betting abstractions are highly exploitable once you figure out the thresholds. I have no idea how to solve this, except maybe to randomize betting sizes so your strategy doesn't assume a 20bb pot ALWAYS means a 20bb pot but could be 15-25bb e.g.

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Cheers.