I need to change my Biometric residence Permit (BRP) to my married name, however my passport with my maiden name is still valid (I renewed only last year). Can I apply for the change of name in the BRP and keep the passport with my maiden name?
No. UKVI will not normally allow you to have different names on your passport and BRP. If you want to change your name on your BRP, you must change the name on your passport first (indeed, if you change the name on your passport you must report this to UKVI).
You will need to complete form MCC.*
The fact that you cannot have different names on the two doesn't seem to be officially stated anywhere, but I know from personal experience that it is true. You could try filling in the form without changing the name on your passport, but I suspect that UKVI will ask to see a passport in your new name.
* Confusingly, the guidance at the top of the form says that if your "current grant of leave was made by means of a BRP" you should not use this form, but should "make an application for a new BRP, using the form for no time limit (NTL) or an application for transfer of condition (TOC)". This makes no sense, and I suspect it is supposed to say "current grant of leave was not made by means of a BRP".