Unfortunately, you are not in the same situation (at least it seems so), as you need to cash cheques.
And:
1) Cheques are a financial instrument that no EMI (TBMK) is able to handle, you need a traditional bank.
2) If the director of your LTD is not resident in the UK then your chances to be onboarded by some UK traditional bank︀ are zero, in fact.
3) Any non-UK traditional bank will probably charge a lot for︁ accepting UK gov cheques and will definitely process it for weeks.
So, either take it︂ as it is; or hire a UK-resident director (if you do not have it) and︃ then go for some UK traditional bank.
Sorry but I can't share with you any︄ more positive news 😉