I had almost this exactly situation happen a few years back. I was leaving a decent position for another decent position because it was closer to what I wanted to do with regard to the projects I'd be working on and was paying about 15% more.

I hand in my letter of resignation, boss thinks it over for a bit before offering me a 30% raise. I respectfully decline and start to leave before she doubles the raise.

This is where I had to think about it - I wasn't leaving for my dream job or anything, just another 9-5, and here I was with 4x the difference on the table. I said if she could have me a 3-year employment contract by the end of the day I'd sign it at the agreed salary, which is when she immediately started hedging about needing to talk to the attorneys and the owners, etc.

TL;DR - Counter offers are bunk.

I was pondering the same strategy, and that same problem came up: the window of "well, I need the lawyers to get into this" and have the process stall long enough to let both deals die and get screwed.

Is it possible to present your own contract and say "have your lawyers look this over and give me an answer in 48 hours"?

I would imagine so, but the downside is how it'd look to the employer. It might be taken as basically saying this:

"Oh, you have a counter offer? I thought you might! I just so happen to have this legally binding contract here..."

Very true. Thanks for pointing that out.

The idea of the contract was that I could stand being in my current position as long as the comp was better. But this course of events would just lead to a lot more downstream pain than expected.

Higher up in this chain there was the mention of asking for a severance package, which seems way more easier to mention off the cuff and not look like a premeditated f-you move.