Cooper,
Sounds like you could benefit from some expert guidance from a good tubist and teacher. Generally, students learn to play on what they get from school and what their teacher plays, out of which you develop a style of playing on specific horn designs you can follow, or at least branch off from knowledgeably. I'd suggest finding good tuba teachers in your area and get some lessons and guidance, if you haven't yet. You might try more than one.
Teachers typically have ideas on the best horns for you to grow with, leads in their tubist network, dealers with whom they have relationships, and deals which might be available that might not yet even be on the market. Often, because of downsizing, collection-adjusting, or no longer playing (or dearly departed), sellers would sooner sell at friends & family rate to a passionate student they meet through a trusted professional than hold out for top dollar from complete strangers, no less go through the effort of managing a public sale. Many of the best horns and deals are brokered by insiders, and never get advertised.
Pros can also test specific horns you're considering, or have trusted testers do so remotely. Every tuba has its own unique qualities, strengths, and weaknesses. Even from leading makers/dealers, there are some not-worth-the-money-for-what-they're-asking tubas out there, or just not right for you, and there are some amazing down-market, used, and modified/custom horns just waiting to be discovered. It's among the unlikeliest, older, most thrashed or Frankentubas are found hidden gems that might be had for a deal. Many tubas are offered for sale by people who really don't know the value of what they're selling -- so it's up to buyers to be able to evaluate the true qualities of a horn: playability, timbre, intonation, durability, fit for your style of playing, size, ambition, finances, etc. then compare that to what's on the market. And based on that, make an informed offer. Often, if you've got a knowledgeable pro helping you who can make the case for what they think a tuba is really worth, sellers who know less will be convinced to accept your offer despite how it may compare to their opening asking price.
The best way to find out is by having a pro who knows you and your playing and whom you trust play prospective horns. And then you should play them, too, before finalizing a sale -- you need to be comfortable with what is ultimately your choice.
$3.5K is a good starting investment for an accomplished high-school player, which can afford you a variety of tubas very well suited to advanced youth/community orchestras, higher ed auditions, and degree programs. If you think you'll apply to music performance degree programs, I encourage you to start looking, and get a feel for where you aspire to go, think you could go, and who the tuba professors are there. They could be another source of guidance and leads to tubas.
After my first few years learning to play on my junior and senior high-school beat up old Miraphone, King, Conn, and Červený 6/4, I bought my first tuba through the teacher I had while I was in high-school (who played with the local symphony orchestra) and then, in senior year, through the tuba professor at the conservatory Saturday pre-college program where I had begun to study. Both got me great deals on hand-picked horns (OMG the stacked ringing overtones and the deal I got on that
Červený Piggy!) which were well suited for me (skill level, physical size, ensembles in which I played, budget, etc). They also brokered sales of those horns to younger students when that time came.