Additional police registration of changing employment after having registered the new BRP?

Dear all,

Sorry for the following lengthy explanation. I have recently received a new BRP due to extending my visa And changing employer. The new employment is in the different city to where I lived (and worked) previously. When I received my new BRP while still living at the old address (and still working with the old employer), I registered the new BRP with the local Police. Now I start working in the new city, but move to a new address only in a few weeks (live in a hotel and/or commute back to the old address temporarily).

Do I need first to register my new employment and then again come to register changing address at the local Police?

I noticed that on my Certificate I actually don't have any entries about previous employers (I had two) - only arrival, new BRPs (I was applying for extension every time when changing employers as I had only fixed-time contracts) and address changes. Is this bad and going to cause problems when applying to ILR, or does registering new BRP associated with the new job counts also as registering new employment?

Many thanks in advance!!