Mesotherapy Vs Liposuction on Someone with Hypothyroid?

Mesotherapy is dangerous and of doubtful benefit. I was asked to speak about it by local news in Atlanta and did a world literature search. I could not find one article demonstrating benefits but many outlining complications.
One of my patients (breast augmentation) chose mesotherapy and came to me with her post treatment problems. She had multiple pitted areas on the hips and flanks, several of which were draining dead fat. The treatment is a year of healing. Her once beautiful figure (with a little bit of hip fat) will no longer be uncovered on the beach.
I have never seen that happen with liposuction after thousands of cases.

Mesotherapy is controversial in the United States. Doctors have concerns about the treatment because it is not yet FDA approved, although early studies are underway by board certified plastic surgeons.
In general, what you are referring to as mesotherapy physicians refer to as "injection lipolysis". Injection lipolysis may be good for very small areas of localized fat, such as that around the belly buttom, the "bra strap" area, upper back, etc. Some of these areas are difficult to liposuction and may be more amenable to injection lipolysis.
Liposuction is more appropriate for larger areas that need to be contoured. Remember that injection lipolysis in not FDA approved at this time.