They will bitch about doing it and that will add a bit more to the cost.
I was assuming that you would be doing the work yourself
Dealers would not know how to tighten their own trouser belt unless there was a service notice telling them how to do it and what new parts they need .
When confronted with one, I go through the wiring diagrams both earlier models & later one to find the last one before the keyless switch and then the first one after the keyless switch and from them I work out which way to go
Husqvarna wiring diagrams are in the front page of their parts books
You should be able to rewire it according to to a YTH 2242 and if necessary buy a new loom which is $ 50 + a key switch which is $ 30
As compared to the keyless switch which is $ 380
If you want the reverse cut out to work then you will need another couple of sub looms / pigtails but again the whole shooting match is under $ 100
However there will be 2 to 3 hours of dealer labour .
Because of franchise rules you may need to take it to a different brand dealer , a freelance mower repair shop or even an auto electrician .