The electrics can be part of the BC involvement or you may need the compliance certificate, often just a tick box with the original application.
The LABC may charge you to do an EICR which you may not get, they are not required to give you a copy, and it would be done by an electrician of their selection, not yours. Some electricians do pay extra on there scheme membership to have the EICR covered by the scheme, but anyone who thinks they have the skill can complete an EICR, it could be done by some one who has only done an on line course.
So as a result the EICR does not show if the home is safe, it should do, but since anyone can write them out, the LABC normally have a list of electricians they trust, and it needs to be completed by some one on their list.
In the main the EICR is done before a fuse box is replaced with a consumer unit, to highlight faults before it is changed, so for example if your central heating has an earth fault, which will trip a RCD, one wants to know before it is put on RCD protection so fault is rectified before the box is swapped.
I did not do an EICR first, and there was a problem that one set of lights was using the wrong neutral, as an electrician I soon found it and corrected, but had I not been an electrician not doing the EICR first can lead to unexpected expense correcting faults. But once the CU is fitted and an EIC is completed it is very little extra to also do an EICR but since not rented personally seems little point.