In connection with the FLR form:-

Q. 1.4 Your full name as in your passport or travel document

This is the name under which the visa and BRP will be issued.

If you change the name on your passport, you must also update the BRP

You can still include your married name on the FLR.
You can use your married or that on your passport & BRP for everything else you wish. The marriage certificate is the key in the audit trail.

My best advice is to keep your ID on all documentation as simple as possible.

The UK is very relaxed about the names you use.
Many folks don't bother to change their passport name until time for renewal.
It's up to you.