Should I use an iframe rather than ajax?

I don't see a reason you shouldn't use iframe. I think using iframe to embed pages inside your website is fine. That is what an iframe is for.

The iframes used to be bad for SEO, however, today web crawlers can usually travel between your website and iframes content.

Using iframe will be easier and simpler for you instead of using "ugly regex". Why don't keep it simple, right? KISS principle

I would say... keep it simple, stupid. Go for the iframe solution, I know it kills a part of your soul, but it is the easiest solution to your problem and represents the lowest change of running into problems in the future.

KISS.