Physically familiarizing with which two orchestral instruments would be a good compromise between having to buy all the instruments and gaining leverage in orchestral writing? My own opinion is French horn and cello; because these two are among the most challenging instruments to write for. I am a composer and have already written a few orchestral compositions, which were performed by a local orchestra. However, I want to enhance my orchestration skill further by actually assimilating myself physically with each orchestral instrument, rather than just reading orchestration textbooks. I don't play any other instruments besides piano. It would be expensive for me to go and buy each orchestral instrument.
I would strongly advise you NOT to buy any instruments. You don't need to. Do you imagine composers play lots of instruments? They don't. The only instrument Berlioz played was a guitar - tolerably - yet he wrote a book on orchestration! What possible use would it have been to Ravel if he had learnt to play the harp?!! If you buy a violin you'll end up writing only what you yourself can play, and you will be an absolute beginner on it!
Composers learn orchestration by listening, by studying scores and by transcribing recordings. When you hear some amazing sound in a piece of music, you look at the score and find out how it was made. If you hear an amazing violin piece, you get the score and study it, learning what's possible on the instrument: not what's possible for YOU.
Composers can also learn something about orchestration from friends who are musicians. Befriend that local orchestra. If they're good, why not ask if you can sit in on their rehearsals? Sit close to the conductor so you can hear what s/he says. (Are they still working in Covid times?)
Transposing is not difficult! Spend a week doing it and you'll master it better than you would by going and buying a trumpet! Of course any music-engraving software will do it for you.
We have it pretty easy nowadays. All of Nadia Boulanger's students had to sight-read scores at the piano! We can listen to almost any piece again and again, free. Berlioz had to go to a lot of concerts and concentrate!
I believe that the instrument you are best served by getting is a professional notation program with good orchestral sounds. The two that I personally have experience of is Dorico and Sibelius. The notation program will allow you, to a certain externt, hear the instruments and orchestration.
Next, get a lot of musician friends. Preferrably musicians that also teach. The reasoning is that you want your music to be performed. In order for that to happen you need to write in a way that all musicians in the orchestra can play the parts. And in order to do that you need to know the level of the orchestra as well of what instruments an orchestra typically has. Even if you write for a professional orchestra, they might not have 4 Wagner tubas easily available, which might stop them from programming your work.
When you have a decision on the "level" you are writing for, say a highschool orchestra or London symphony orchestra, check with musicians how difficult the parts are to play. Every instrument has things in the orchestral reportoire that is on a technical master level -- you want to avoid that level unless it is absolutely required for the kind of expression your music is aiming for. Check with the musicians: maybe that interval is difficult, maybe that long tone in a wind instrument simply is too long and so on, maybe that run is too fast. Try to find those places by talking to musicians and make the parts simpler unless the effect for the listener is worth the effort. (A professional player might spend months learning a difficult passage when needed).
A French horn and a double bass - that's the answer.
But it's a moot point, because the presupposition of the question is wrong. No one in the history of music became a musician, or a "better musician" by "acquiring an instrument".
(since this answer got got both upvoted and downvoted, let me justify the selection: these two are instruments which fit OP's constraints, are no worse than any other selection - given that we have no criteria at all. But the most important advantage of these two is that they are very different than piano, and have a relatively short time from picking them up for the first time to playing parts which are close to what's actually played in typical orchestrations) (ok, a triangle would also fit the bill, but getting a triangle is probably not what OP meant by "acquiring an instrument")
I believe you should satisfy your curiosity, but you might stop short of purchasing the instruments. Perhaps you could borrow or rent a larger variety, so that you have some familiarity, and then as composers do (and @OldBrixtonian wrote), you can consult with genuine experts on those instruments when necessary.
Having some direct knowledge of strings, winds, brass, and percussion isn't going to hurt you and will perhaps engage you in further exploring those instruments compositionally.
As an alternative (or in addition), you might consider trying some more "exotic" instruments, not part of the orchestral canon. By taking yourself outside the more traditional sounds, perhaps you'll discover creative uses for both traditional and non-traditional.