You most likely do risk invalidating the insurance policy. Such indemnity policies typically have clauses to say that you mustn't carry out any action which alerts the local authority to the situation. The risk from the insurer's point of view is that if the local authority are alerted they may decide to take enforcement action against the earlier breach, and the insurer doesn't want to be liable in that scenario. You should check your policy wording to be certain.
Enforcement limitation periods are as follows:
In practice local authorities rarely enforce breaches older than 1 year (presumably because of the increased risk/cost of going to court vs simply serving a notice), but of course you can't rely on this being the case.
As an alternative to applying to the council, you can use a private building control company (an "approved inspector") provided that you instruct them before you commence any works (they are not permitted to issue retrospective building control approvals). They are not going to care about the previous breach. The local authority will still be alerted to the fact of the new works but this is just a short notice sent by the approved inspector at the start and the end, and the local authority are far less likely to inspect (it has never happened in my experience). There are plenty of such companies; just google "building control approved inspector". But in any case you may still risk (a) invaliding the insurance and (b) triggering enforcement action, it's just that the risks are lower.
You probably will be required to submit a party wall notice. If you can gain the cooperation of your neighbour, it is not necessary to go through the formal process of reaching an agreement, and they can simply reply to your notice to say that they are happy for you to go ahead. If they are not willing to do so then it will become more complicated and you will probably need to appoint a party wall surveyor to reach an agreement. This will boil down to a bit of hassle and money, but isn't a big disaster otherwise. Be wary of party wall surveyors sending you marketing which makes it sound like you need to use them (quite common after you do something semi-public like apply for planning permission); always try to settle it with a written consent before you waste money. There are plenty of online resources and books which can guide you through the process.
Although you didn't ask about planning, you should be aware that pursuant to the Town and Country Planning Act 1990, Section 171B(1), you gain immunity from enforcement 4 years after completion of the works. You are also entitled to request a Lawful Development Certificate for Existing Use from the Local Authority to confirm the immunity, although it is not necessary to do so and it is perhaps not recommendable in this case as you might alert them to the building control issue and you would almost certainly invalidate your planning indemnity policy (albeit that would be less important than the building control policy).