Interestingly, while searching to see the discrepancy between mining pools and GBT, I found this site by 0xb10c:
https://miningpool.observer/template-and-block. It definitely doesn't tell the full story due to a myriad of factors affecting the selection of transactions but some of the pools are definitely selecting transactions themselves: High discrepancy between GBT and the actual mining pool, F2Pool has an average of 99.95% compared to the rest which averages around 99.85% with regards to total capacity.
F2pool, Ocean pools are going to be outliers here, simply put, F2pool censorship transactions for political agenda, Ocean censors ordinals/BRC20 tokens, mempool has a similar and better looking indicator called health, here is an
exampleOther pools that have acceleration services like Viabtc will also have some bad selections for obvious reasons (I talked about how Viabtc scam miners in the past), you also have propagation delay issues, so the perfect set of transactions isn't always the same on all nodes, your node may be seeing 10 good paying transactions that mine doesn't see at the time of solving a block but you would think that I missed on adding them because my LP solver is bad.
I don't know if any pool uses LP solver post/prior to getblocktemplate, i'd assume the safe option is to do it before and not after since getblocktemplate will do the verifications needed to ensure your block gets accepted by the network, your LP may add a transaction that would render your block invalid, I doubt that any pool is willing to risk a whole a block just to add a few sats to a 6-7
BTC block, but as you mentioned earlier when fees make up a majority of the total rewards it would be worth the effort and the risk.