It’s not rude to count it. It doesn’t need apology or explanation.
You’re engaged in a business transaction. If you were in a shop and someone gave you £200 in £10 notes you’d count them out.
The person will not feel awkward and neither will you unless you get unnecessarily flustered.
Simply say ‘Thank you. I will just check this in front of you’ and proceed to count the cash out loud ‘ten, twenty, thirty,’ etc. Finish with ‘Great. Thank you very much’
If you get flustered or confused when counting, just say ‘Oops, will just start again’. If there seems to be too little or too much, say ‘I will just count that again’.
People make genuine mistakes, they try to be crafty, they overpay mistakenly too.
You do t want to be contacting them after they’ve driven away and they don’t want to be hearing from you again after. This kind of deal is done in person and without later recourse, so it’s vital it’s done properly for the benefit of everyone in a way everyone can see and so there’s no doubt or confusion.
If you don’t like dealing in money, don’t buy and sell like this.