Congratulations on your purchase of a SIG P226, docker01. It's a sweet and reliable gun in any version and I'm convinced you'll love it for years to come.
As for the answer you dealer gave you, well your dealer is plain wrong and was probably trying to upsell you to a 9mm.
I own P226s in 22LR and 9mm and I go back and forth between the frames and slides without any issue.
Check out the first question/answer on SIG Sauer FAQ:
"Centerfire Kits (9mm,.357 & .40) are intended for use on .22 Classic Framed Pistols."
Source:
https://www.sigsauer.com/support/faqs/
You do not need to read further if you are just looking at converting your 22LR to 9mm. You can and that's precisely what SIG says the 9mm kit was made for.
The rest of my post is just a personal rant about something else SIG states on their FAQ.
SIG's answer goes on to recommend against using a 40/357 SIG conversion on a 9mm frame, and in the same breath they're saying it's OK to run these conversion kits on a 22LR frame, which makes absolutely zero sense. If the 22LR frame is strong enough to accept 9mm, 40 S&W and 357 SIG conversion kits, there is no logical reason why you couldn't put a 40 S&W/357 SIG conversion kit on a 9mm frame as well.
I think the bean counters at SIG Sauer are trying to upsell 9mm owners because they make more money selling a whole new 40/357 gun instead of a just a conversion kit.