Lots of choices for that...personally I default to a 1/4" upcut end mill when doing a thicker cut like that...Upcut to help keep the cut clear of chips/dust...I use Amana bits almost exclusively...to me it's like putting a plastic lens on a Pro Camera to use cheap bits on a multi-thousand dollar machine...wrong place to be a penny pincher ;-)
The problem is: He wants to do a full profile cut through 2" of oak. Cedar and redwood is a lot easier to cut than oak. The few times I've tried to cut through 1-1/2" birch or oak it has been a major pain. Increase that to 2" and the problem is exponentially increased. The cut could be made a lot easier if it were done 2-sided, cutting halfway from each site. Or cut (2) 1" pieces and glue up.
You can use Amana 46577 1/4" 2-flute 1.5" DOC end mill with multiple passes to get the full 2" depth. The key to using such a bit is that in addition to using multiple passes you'll have to also make the groove wider than the cutter to have the chips some place to evacuate to. I'd probably do it with 3/4"-1" for the first 1/2-1" down and 1/2"-3/4" wide from there to 1.5" down. Following with a vacuum point may also be necessary but you can make some nice chips instead of dust with oak so they tend to evacuate better.
I had a similar task with these - they're 1-3/4" thick maple. I used a 1/2" end mill because it had the longest cut depth I could find (1-3/8"). I also did more than just a profile cut and made a .6875 wide pocket to give it room for the chips to evacuate and made 1/4" deep passes with it. It worked but I probably wouldn't do it again. The 1/2" end mill is a lot to ask of my AR8Pro - it's MUCH happier with a 1/4" end mill.