The classic "entry level" horns are the Selmer Bundy and the Leblanc Vito - about as cheap as you are going to get new, and for one with casual interest, good enough to get by. Another option is the entry level Yamaha, but I can't offer an opinion on same, as I have only seen one of them in the flesh, and have never played one.
Used instruments open up your options somewhat, but also put you into the realm of the school owned instrument. Bass clarinets, like baritone saxes, get beaten all to hell when in scholastic use. Personally, I would avoid them like the plague, but others will certainly differ on this one. (I only buy new cars - same abuse arguments, too expensive to take a chance on buying used.)
The one benefit of going for a used instrument is the cost factor. Deductions of 50% or more are not uncommon for most musical instruments. Just be sure to take advantage of the 100% guarantee policy when you buy (if one exists).
For many years, Selmer was the go-to brand for a quality pro bass clarinet, either standard (to low Eb) or extended (to low C). (This accounts in part for the large number of pro Selmers on sale on eBay.) I own a number of basses, but only one to low C. If I had to buy my first one these days, I'd skip the extended range and spend the money on something a bit less frivolous. Again, others may differ here.
Buffet has always made wonderful instruments, but the Buffet rep wouldn't deign to send me a group of them when I bought, so I've not owned a Boehm Buffet, ever. Albert Buffets are another matter, although few would want to learn the Albert "system" these days, and the only ones you are going to find are going to be very used.
Leblanc have been beneath contempt with me for many a year, but I understand that they have shifted to the "long rod on back of horn" style of register key in recent years, so that may no longer be the case. If presented with a "non long rod on back of horn" Leblanc, even a professional quality one, run – don't walk – away, for those horns are dogs. And, of course, even with the newer model, you have to worry if they are still producing them...
Boosey Hawkes made good basses many years ago. I've not seen one on sale since the advent of eBay, and I doubt that many other have.
There are Chinese basses on the market. The only ones that I have seen were poorly built, out of tune with themselves, and had keywork that was way too "bendy'. There are also those who claim quality for Chinese instruments - all I can say is that I've not yet seen one that measures up to my standards.
I would also avoid horns without an integral floor peg. As you are interested in jazz performance, as likely as not you will be playing standing up. You can play bass from a peg while standing if you rest the foot of the peg on a chair or stool. (I do this when performing the classic Sentimental Journey arrangement that uses bass clarinet, rhythm section and female vocalist. I always get questions about what kind of saxophone I was playing...)
Mouthpieces? I've always (for over fifty four years now) played on Selmer stock pieces. Not the crappy ones shipped with the student horns - I mean the signature Selmer models with the grading system of letters and stars (i.e., C** or D or G or H). I prefer the HS** facing on sopranos, but I've not seen these for the bass. A more open facing D, E, F, G or H combined with a 2 1/2 reed would be best for someone playing in a jazz group. A harder reed and a closer facing makes for better upper register action.
Others will tout the Grabner or the Bay mouthpieces. I've only tried Charlie Bay's, and I didn't find enough there to want to change. Don't know squat about Grabner or Vandorn.
As you play jazz, you might not be offended by the angle that the student horns present the mouthpiece to your gob. However, you should be aware that Charlie Bay makes a modified neck that converts horns (both student and pro) to a much more clarinet-like angle. Iffen you take the mouthpiece socket from the Bay neck and combine it with the neck socket piece from a Selmer horn, you get an even better mouthpiece angle.
Clarinet embouchure is different from sax embouchure. If you taught yourself to play a clarinet, I'd recommend a few lessons from an experienced clarinet player until you learn all of the differences as to how your mouth interacts with the clarinet mouthpiece.
Ligatures vary. There is the classic metal model, a few aftermarket options like Rovners, and the one that I produce. For a casual player, just stick with the one supplied with the horn. The horn makes up perhaps 20% of what you produce, the mouthpiece, reed and ligature combination perhaps 15%, and all of the rest comes from north of the mouthpiece, i.e., you.
