Is it possible to pay transaction fees from a different source?

Yes.

The token sender is determined based on the first transaction input, while other arbitrary inputs can be added to a transaction.

This allows to craft transactions, which use an input to tag the token source, but other inputs to pay for transaction fees. The token sender likely has a tiny fraction of bitcoins anyway, because he or she received the tokens in the past.

It is possible to use the raw transaction API of Omni Core to craft such a transaction manually. As general guide, see the following tutorial:

• Use the raw transaction API to create a Simple Send transaction

To create, sign and broadcast funded transactions, in a similar fashion like omni_send, there are two RPCs:

• omni_funded_send
• omni_funded_sendall

Please also see this introductional post about transaction funding:

• Medium: Introducing funded Omni Layer transactions or “how exchanges can save money”