I'm not expert on PSBT, but
1. BIP 370 (PSBT v2) improve BIP 174 (PSBT v0) which add more feature, mainly adding more input and output.
2. TX fee actually determined by subtract total coin on the input and total coin on the output. That means determining who's responsibility to pay TX fee entirely depends on all inputs and outputs.
Secondly in the usual cases the party providing the fund usually bear the transaction fee but you can also distribute the fees between the two party by using
output tagging (BIP-68) or you can negotiate the fee distribution outside the protocol
BIP 68 isn't about output tagging. It's about timelock which determine when an UTXO can be spend, not about managing TX fee. Here's simple explanation about BIP 68 to other reader.