I'm far from convinced that it's a good idea to connect a charge point to a spare way in a CU, not least because it still means having a separate enclosure to fit the required DP Type A RCD/RCBO anyway, so it makes more sense to just supply that separate enclosure from the incoming supply. This also avoids adding to the heat build up in the CU from the MCB being run at 32 A for hours on end.
It's also not possible to connect to a spare way in any CU that has an RCD, either, as you cannot daisy chain RCDs for priority reasons, plus there's a risk that the RCD in the CU (which will probably be a Type AC) will be blinded by any DC component from the charge point when in use, so reducing or removing protection for the whole house (or that part of is fed from that side of the board if it's a split board).
No need for separate BC notification if done by a Part P electrician, as the EIC will just be electronically lodged on the database. However, it sounds as if this installation may not be compliant with the wiring regs (specifically BS7671:2018 Section 722) and so it should not have been signed off, perhaps. The signature block on the EIC includes wording that the person signing it (even if that signature is electronic) has inspected and tested the installation and ensured that it complies with the regs. Any non-compliances have to be listed, with reasons given, but failing to comply with the requirements of Section 722 would not normally be allowable for a new installation.