The way I would approach this is to take the offer from the competitor.
Assuming it is, as you said, a compelling role with a comparable benefits package, then accepting the large signing bonus is a win-win for you. It's important that in your career, you focus on yourself and your needs first. While money is not always the most important factor, you've clearly communicated that it's not the only factor in your decision. If it's the only differentiating factor, however, why not reward the company who showed initiative while benefiting yourself?
However, I'll argue there are more benefits to taking it. By working for another employer, you will vary your experience, making you more valuable. You'll get the opportunity to learn different skills, and communicate and operate within a company that will -- at least in some ways -- be different from the one you currently work for. You'll get to see how a different culture thinks and approaches problems. You will also get to be the outsider with a fresh perspective, which can help your new employer and boost your reputation.
You probably don't stand to lose much, either. Only self-destructive companies hold grudges because someone took a role elsewhere. If you decide to return later, or the company begins a project you're eager to work on, you can most likely return. On top of that, you're a specialist in an under-served area, so they'll probably still need you just as much. Plus, you'll have spent time improving your skills somewhere else, which didn't cost them a dime! Unless you are on a pension plan you'll lose (rare these days), your benefits probably won't suffer from changing companies for a while.
As someone who has taken a counter-offer to stay, I can tell you there is very little benefit from that choice, even if there are few (or no) downsides. But there are generally downsides, maybe even some you can't predict. You'll also always kind of wish you had checked out the other company from the inside. Even if I had returned a year later, I would have benefited much more from taking the new, exciting role, and having more negotiating power on my return (maybe even receiving a signing bonus).
One last thing to remember is that we're generally scared of "the unknown" or change, but we're also bad at estimating risk. In this case, accepting the offer is low risk, but comes with a good chance of reward. I'd argue that seeking a counter-offer from your current employer is actually higher risk than taking the one on the table. What if you're seen as disloyal and money-seeking by your current employer? What if you also end up losing the new offer by waiting too long, or because it gets back to someone who takes offense that you're using it as leverage? Granted, those are small risks, too, but probably more likely than accepting a lucrative, exciting offer will damage your career in some way.