How do I use a second job offer to improve the lowball offer for the job that I prefer?

You're overthinking this by a lot. And you are making the mistake of starting from the view that they will never cover the "gap." So this means you have accepted their premise that their offer is a legitimate one. You can always split the difference between two reasonable offers, but when one offer is out of whack, then you have to reject that offer, period. You must start with the premise that their offer has no standing, any more than my offering to hire you for $10/year. If you are psychologically prepared like then, then here's what you do:

You go to Company B and tell them the truth: I am passionate about your company and this job and would love to work for you. But your offer of $100K doesn't reflect my actual market value, $150K. I have a standing offer for that from a company of comparable revenue and size. I'm not interested in a long back-and-forth and I'm sure you're not, either. If you truly can't do this or don't feel that's my value, I respect and understand that, as I'm sure you'd respect and understand why I can't accept your $100K offer. [Note: I can't emphasize enough how important this last sentence is. You diplomatically reject their offer completely. This is the diplomatic equivalent of walking out of the room, but very gracefully. If you deliver it right and you believe it, it's tremendously effective.] But I do hope I can work for you -- what can you do? [Then, stop talking. See what they say, no matter if the silence is 2 minutes long. And, btw, are you saying this to HR? I hope not. You're saying this to your hiring manager.]

Further points:
-Do they have an amazing bonus or stock package?
-Are there other reasons the offer is lower? I'm not taking those into account.
-If they say: How about $125K? You MUST respond, "I appreciate that very much, but I believe my market value is fair and proven, so I would have to say, "No, thank you." Then, again, be silent.
-Other alternatives:
-They pay you a $50K signing bonus and guarantee in 12 months to raise your salary to $150K.
-Or you accept the $100K, with the understanding that they will review you in three (3) months and move you to $150K then, if you're not fired, of course.
-Or they give you a ton of stock or a 100% bonus plan. But first they have to accept the market value itself.

Also, consider the possibility that they are cheap and simply do not value anyone where they should. I don't know enough details to answer any of these questions, but this is how you should approach it.

A few thoughts on this, that might help you in your decision:

1. I would challenge your assumption, that "the jobs are otherwise similar". Something must be different between the jobs, so start off somewhere else: Where will you learn more? Where will you be the happiest in your everyday life? If it makes you miserable to leave 50k on the table - which would be totally fair - then go for A.

2. If B has a valid reason for the lower offer, or they can offer you something else (bonus possibilities, profit sharing, options, education), then go for that.

3. Thinking experiment: If you offer B, that you would work for them for 150k, and they say no - would you feel happy working for A. Would that be equally interesting?

33% below market level is a lot, so I would understand, if accepting such a low offer, would create some negativity in you. If B can not offer higher, it takes some greatly interesting future opportunities from them to make up for the lost salary. So no matter what, ask B for more, but be ready that they could say no, and take their offer away.

Given the information I would say, that it is really a question of your preferences. If you would like to discuss it more, so you become more aware of your preferences, give me a call.

Good luck with your decision.

Best regards
Kenneth Wolstrup