I had this problem initially. I don't have tennis elbow but I noticed that I was having the symptoms after long rides. And I too used to ride MTB exclusively, and I ride MTB with a much higher handlebar than seat, my setup is very much like a BMX setup, with a slammed seat.

I ride hardtail, so I am use to my legs having to absorb a lot of shock and they are good at being loose when they need to be loose, so that transitioned to the road just fine. However my arms were not used to being loose, since my front shock handled that entirely. When I went to a road bike I didn't loosen my arms up and the shock just went right into them. for me the quick answer was to ride in the drops, that way my arms have to be bent, and I am now beginning to learn how to absorb shock with my arms the same way my legs do it without me even having to think about it.

So my advice, try drops if you can until you get it down.