Is There a Definitive Way to Determine if Crown Fits Properly?

Taking an xray of the crowns should help, as well as a dental explorer (the metal pick that dentists use all the time). If the xray looks great and the explorer feels smooth, then the problem is likely something else. Try dipping toothbrush bristles in hydrogen peroxide and scrubbing the area 2 times per day for a week and see if things change.
A $4000 MRI is not really a good choice here, and the details given aren't precise enough to determine fit.

Well done porcelain dental crowns ought to settle in and last for years with no more than minor issues. We always check the fit of any new crown we fit with an X-ray. Your dentist is correct that not all issues with the fit show on X-rays, but it is still the single most useful test to verify the fit. There are no other types of X-rays or MRI's that are helpful or pragmatic.

Often all porcelain crowns are actually bonded. This adhesion process is very technique-sensitive. If the edge of the crown is under the gum line, this can affect the adhesion process. Sometimes the gum irritation is not from a bad fit of the crowns, but from some extra cement left behind. This does often show on an X-ray. 

The most practical way to proceed right now is to request an X-ray. Check both the fit of the margin and check for any excess cement. You may need some time with the hygienist and possibly even be numbed so she can do some additional fine cleaning there. In our office our hygienist is trained with the laser. The laser is sometimes an exceptional tool to help improve very fine or delicate gum problems.

 

Whether the crowns are a good fit or not BEGINS AT THE BEGINNING. When the dentist FIRST begins tooth shaping the margins must be placed so the biology of the gum tissue (gingival sulcus) is not compromised. Go too deep and the gum tissue will always be inflamed after the crowns are placed. A crown can be a good fit but be too deep in the gum and it will always be red and puffy.  If the preparation (shaping) of the tooth doesn't result in clearly defined margins which are recorded in the impression the crowns won't fit correctly. Then there is the responsibility of the dental lab. A good lab can't make well-fitting crown with a flawed impression. An inferior lab can't make a good crown from a good or bad impression. In the end it is always ultimately the responsibility of the dentist. Determining whether a crown fit is good is pretty simple. When in doubt it's always okay to get another dentist's opinion.