The only reasons why you need to be connected to the Bitcoin network are: 1) you need to check if the funding transaction has been confirmed  2) you need to process new incoming blocks and check if the other party didn't broadcast a commitment transaction 3) the other party might request a fee update to the latest commitment transaction; if your response is too low or too large then your peer might close or temporarily disable the channel

Currently, the third point enforces connection to some Bitcoin backend, but I could see someone modifying their client to use some third-party source for fee estimates. However, this would put them at a risk as someone could close the channel and keep using it. The modified node would keep singing new commitment transactions even though their inputs could have been already spent which would make all of those transactions invalid.