Thinking about the sound is good, but there are some more things to think about.
Would like a stand (bass fixed even if your body likes to move) or an endpin (or an angled foot).
What kind of strings do you like to use?
What kind of pickups?
Has it active electronics (important if you want sound control and a piezo)?
Better get double bass tuning machines than bass guitar tuning machines on the EUB.
Get an ebony fingerboard or similar really hard wood for it.
Check the body separator if you want a endpin or angled foot instrument, also with the bow. You should keep the instrument in place without rotation. (If the surface is to even it might slip, but winding something around might help, I use rubber foam tubes.)
So there is much more to think about than only sound. A good instrument that could be hold and played well is more important than the sound. You can always put a pickup on it and change strings. Try as much as possible before you decide what is good for you. Otherwise you might buy and sell EUBs a few times more than needed. (And this will most probably kill your budget.)
For your budget I think a good used instrument might be the best choice. Probably with bass guitar tuners and being nailed down on steel strings (because of the tuners).
For most EUBs the strings need to be cut or you have a lot of string ends peeking out of your headstock, that's the reason for the steel strings.
If you can get a used Yamaha silent that might not be completely out of your budget and not a bad choice.
