A third dose is different from a booster - depending on the biologic and whether he's on MTX, he may not have made enough antibodies to the vaccine or have enough T cell mediated immunity. The third dose guidelines for immunocompromised individuals specifically mentioned those on anti-TNFs or an anti-metabolite drugs like Methotrexate. They do not have the same immune response as people not on those meds, so the third dose is necessary just to get them as protected as a healthy individual who gets two vaccines.
Biologics like Entyvio and Stelara are much less immunosuppressive, so whether a third dose is needed for patients on those drugs depends on what their doctor recommends.
But for those on an anti-TNF, a third dose is definitely recommended. That is different from a booster.
There is evidence that protection provided by the Pfizer vaccine (which has 1/3 the amount of mRNA as the Moderna vaccine - which is the one that caused myocarditis mostly) wanes quickly - by 3 months, it's something like 77% effective and at 6 months, efficacy is far lower. The efficacy of the Moderna vaccine is still around 93% even 6 months after the shot.
Hospitalization in teens is rare but still possible, particularly in immunocompromised teens. But even mild COVID can cause "long haul" symptoms or other issues. I have a friend with Rheumatoid Arthritis who got COVID despite two vaccines and developed Lupus as a consequence. She used to run half marathons. Now walking up a flight of stairs is hard. Her 17 year old son with JIA and Crohn's also got COVID and while his symptoms weren't terrible, they led to a major arthritis and Crohn's flare and now his Stelara is no longer working and he has already failed Humira and several other biologics, so his mother is very worried about what they can use to control his diseases. He had COVID 6 months ago and is still flaring, despite increasing the frequency of Stelara from every 6 weeks to every 4.
So even mild COVID can cause major issues. Your son has had COVID but studies seem to indicate hybrid protection is best - antibodies from the vaccine as well as from the infection. Does he have enough protection from the vaccine? That's something only your doctor can answer and I'd encourage you to discuss this with your child's GI.
FWIW, my nephew got COVID in the summer (he is a young adult - 26 years old) and he was told to get a booster (since he's not immunocompromised, it would be a booster and not a third shot) and he was told to wait a couple more months before having a booster. But then the Omicron variant started spreading and his doctor is recommending a booster ASAP.
Please talk to your doctor - your son's GI would be best,