You only get the baby "firsts" once.
Since you said his "pleas" have gone on deaf ears at his company for less travel, it sounds like he is wanting to stay home with the baby.
Here is the deal:
You said he has the opportunity to climb the ladder really quick at his current job. How solid is that? Is that almost a 100% guarantee? Because tripling salary (but being gone all the time) would be better financial security for your family.
If it were me, I would at least interview with the new job. It doesn't hurt to interview and explore your options. That company may have other perks he doesn't realize. Maybe the advancement opportunity could be there as well?
If it were me, I would interview. Weigh the finances and see if you can make it work comfortably (which you say you can). If so, I would take the new job. I would want to be home every night to see my kid grow up- not through facebook posts and skype.
Just my opinion.