I went through this same problem a while back and hence chose my username because of it. I made a custom neck out relatively heavier wood but I like the look and feel of it so much, I had to compensate for the neck dive.
Try this experiment:
Assuming you have a 4 string, completely remove the strings and the D and G tuners from the headstock. Now play as normal, standing, sitting, etc, but with only the E and A. Does the bass balance well? You should feel a big difference. The Ultralites are about the half the weight so that's how your bass will balance with 4 of them on it. In addidtion to Hipshots, check out Gotoh GB640s if you want the old style but much less weight than stock tuners.
As far as bridge recommendations, I put a Hipshot A brass on my Jazz which helped with the balance. Although the extra weight was nice and the adjustablility and the string changes were faster, I thought it had a little too much sustain (which is good if that's what you're after). So I eventually opted for an original 70's bridge and covered it with an ashtray cover with fish weight on the underside covered by copper shielding.