Would anyone know if bidding for a business class upgrade would be better in terms of saving more money than paying for an upgrade at the check-in desk?
The answer depends on the bid quantum and the delta in the paid fare vs available upgrade fare.
Intra Europe the difference in full fare economy and full fare business is usually under 50 EUR. Whereas the bid-for-upgrade rate might be 100 EUR. Of course if you paid deep discount economy it could be a good deal.
On the other hand repricing a deep discount economy ticket on London-New York on the day could be (from O to J class on BA) over 4000 GBP, but if "airport upgrade offers" are available then they will be about 1000 GBP each way, as I recall, so in that specific case, it might be a good deal, depending on your resources.
If you are asking, "How much should I bid for an upgrade in an auction with limited seats, given prices X, Y and Z?", it sounds like the sort of question they ask at a quant analyst interview for a small hedge fund. There's an answer as to what is logically reasonable but really it depends on how much you value the upgrade.