OK, I'll throw out some things to consider... I'm going to assume that your son has expressed interest in playing at the next level (college) but we both may be getting ahead of ourselves a bit.
He is getting to that age where it's time to introduce him to Strength and Conditioning training along with Speed and Agility - mostly because you need more things to try to squeeze into his schedule . Study up on those things as well - there are plenty of posts you can search here as resources. You'll need to figure out where he is from a physical maturity standpoint. Some 14-15 y.o.'s are further along than others.
There are two schools of thought regarding throwing year 'round. I think the general consensus, as well as the expert Doc recommendation is to schedule a few months off each year. If you choose that route, those few months are a good time to focus on conditioning, hitting and other interests.
Pitching and Catching - there are also plenty of posts to search on this topic here. Some can pull it off but it is a very tough combination. There are usually different throwing mechanics involved. There is a much larger opportunity for the arm to get overused. It is difficult to balance the workload required by the HS program. As a P, you are asked to throw regular bullpens. As a C, you are asked to catch regular bullpens plus go through all the defensive drills and throws. Catching drains the legs, which makes it tough to then pitch. On top of that, you may be trying to squeeze in private lessons for one or both and there are more throws involved for that. In games, those are the two positions that require the most throwing and if you are asked to do both it can be very taxing. Also, if he decides to pitch and wants to play at the next level, he is almost assuredly going to become a P only and give up hitting in college, if not in HS. Is he OK with that direction? Plenty of good position players with strong arms make that transition later so don't worry about rushing into that unless he really favors pitching to playing another position and hitting.
Regarding his build - If he is projected to be quite tall (6'3" +), this is certainly looked at favorably from a pitching standpoint. However, the fact that he is lanky now is a non-factor. He can certainly bulk up over the next few years if that becomes the goal. Good catchers are desired, regardless of height.
Now, back to your list of options...
He has the option to :
1. Continue travel ball and play as well as continue lessons
2. Private lessons only (hitting/catching)
3. Rest arm no throwing. Just work on hitting.
Playing more competitive ball is always advantageous for young players but you have to weigh this with other interests and objectives. Of course, grades/studying always must come first. He may play other sports or have other interests and those are usually good things for a teenager to experience. Then, after he takes care of other responsibilities at home, what is there time for and what is priority? If you buy into the "couple months rest" for the arm, this may be the time. Just make sure he leaves himself enough time and uses a good program to properly ramp back up prior to HS tryouts/practices. Also, remember that catching and/or pitching lessons require throwing so maybe he uses Sept/Oct as time for lessons and then shut down the arm November and December. Start mapping out a plan that covers everything. It only gets harder later when you have to start juggling fall/winter showcases and college camps.