A) Hard rubber mouthpiece mean NOTHING. You need a piece with a clean facing. On bass something hand faced is almost a must. Mass produced mouthpieces are often made poorly because of the small market. A forbes debut does the trick for $30. If you want something really nice invest in one of David McClune's mouthpieces. He makes one especially for doublers and students. It will last you a lifetime. Your mouthpiece could be causing your issue. Don't get something with too wide a tip. A clarinet is not a saxophone, this adage is especially true when talking about mouthpieces. You need something to press up against while your still learning on bass. A lot of people will tell you to get a Vandoren B-50 or WIDE selmer. It's not a good piece of advice for someone starting out doubling.
If your mouthpiece came with your horn, use it for firewood.
B) (Likely) Your octave mechanism might be out of adjustment. This is a common issue on all bass clarinets, and especially so on cheaper models.
C) (Likely) The horn might not be venting enough because of pad thickness. Pads that are way to thick are often fitted onto these cheaper horns. They prevent proper venting both in the octave mechanism and the body of the instrument. This is less of a problem with the lower register, however when your heading into the stratosphere you need the body of the horn to vent properly. It might also just be leaking.
D) More-so then sax, playing the upper octave on a clarinet (especially bass) requires changes in embouchure and air pressure. Practice your clarinet chops to get rid of the ubiquitous sax doublers loose lower lip. On bass a good exercise is overblown 12ths. You take a low note, hit the octave mech, and then transition to the same not above. What do I mean exactly: finger a low E, Hit the octave mech to go to the B, and then release your fingers to go to the E above that B. Go up a half step and repeat. This should be completely smooth with no cracks. Do this until you are hitting your "ubber D" which is everything open. This will take time.
E) Wood/Low C basses are overrated. I won many an audition on a selmer bundy with a Bay neck and an old student brilhart mouthpiece. Expensive basses are only necessary when your milking that last bit of TONE out of your horn. That said I do currently play on a vintage wood low Eb selmer and a Buffet prestige low C, however I have clocked many hundreds (if not thousands) of hours on bass over the years.
F) Do articulation etudes. They matter. The best ones I've seen can be had for free from Michael Lowenstern's website. (earspasm)
G) Your reed might just be bad. Vandoren is notorious for inconsistency in bass reeds. Try Bass Clarinet Rico Reserve Classics for a similar sounding, more forgiving, and more consistent alternative. Don't play synthetics. They are fine on sax, I don't know why but in my experience they always sound horrible on Bass. I believe this has something to do with differences in facing, but I am not an expert. Take me at my word when I say invest in a good reed case and refinishing plate (if you don't already own one). On sax this is (in my experience) a non-issue, but minor warp in reeds on bass make life horrible. Plus, as an added bonus, your reeds will last longer.
H) Some will say your issue stems from a lack of a "true" double octave mechanism. Frankly this is an excuse. You can (and should) sound great on the horn you have.
Hope that helps.