Thank you all for your advice and perspectives. Some people mentioned negotiating further/harder, but the fact is my industry is constrained a bit by the US government and, in my new role, I am pushing the upper end of what any reasonable person would give someone of my age/experience. Unfortunately, skills/ability isn't much of a factor when the government pays based on years of experience and degree(s) obtained. Of all my known near peers and acquaintances, I am now the highest compensated by a decent margin.
Here is the follow up:
I took the outside offer and resigned from my old company. My immediate supervisor and mentor made efforts to get me a 20% raise and a promotion but, in the end, it was futile. Off the bat, the company absolutely would not approve a promotion (They cited policy and time in grade. It's funny because 4 months from now, in 2020, it would not be a problem???). Even without the promotion, I might have stayed with just the raise, but it ended up going 3 levels above my supervisor and getting denied by a VP due to the "current financial situation." (Yet, we were recording record revenue, etc.)
I can't say I'm surprised by the behavior of a large corporation, but it's rather astounding to me companies that act this way function or make profit. I understand there is a desire to remain competitive and everyone is replaceable, but when absolute dimwits who do nothing are getting paid what I was asking and multiple senior leaders are vouching for me, I can't fathom why you wouldn't pay the market rate for a valued employee.
But, I guess that's not my problem...
So, on to the new company and I'm always welcome back at the old company. The old company hasn't come asking for a tuition reimbursement back pay yet, but I assume they will want that eventually... But, there is hope they'll just forget about it.
At the new company, I do in fact have overtime authorized for the foreseeable future so that helps shorten the break-even time on the tuition reimbursement. All-in-all, I'm making about 18% more overall compensation and the promotion sets me up for greater/faster growth in the future to make even more money.
It's way too early to make any judgement about the new company but, no red flags yet...
In conclusion, a happy ending overall.