This could be batted around almost endlessly because we don't have enough facts.
As regards the PCN:
To whom and where was this and subsequent notices sent? You have to find out, we can't. You could ask the authority - and include dates for when they were posted - but do not ask for copies because they'll baulk. As you're asking for personal info which they hold on the addressee you might be asked to prove you're that person and that they may release the info. This might be a barrier, in which case you need to submit a simple, straightforward(despite the name) Subject Access Request to the council.
https://www.lbhf.gov.uk/councillors-and-democracy/data-and-information/your-data-rights/your-data-rights-how-exercise-themYou shouldn't have to explain, but sometimes it's better so you want the addressee, addresses plus dates of posting of:
PCN
Charge Certificate
Order for Recovery
The warrant of control
(you already have the Notice of Enforcement and notice demanding the enforcement fee, yes?)
There's often an issue around the address on the warrant which, if it is made out as per the council's notices, could give the 'wrong' address. This means that the bailiff could not enforce against your premises but could still do so against any car that you own if parked on the road.
Which itself would beg the question why, once they discovered your current address, did they not notify their client to get the warrant reissued with a correct address?
So many unknowns.
OP, over to you. I agree that paying now is the best interim step because this doesn't prejudice your position as regards scrutinising the council's and bailiff's procedures or as regards getting the process reset to the PCN.