Paul - YOU CAN DO IT! One of the challenges of 365+1 is coming up with unique solutions to getting on air every day. Fun!
Let me second Matt's suggestion about TeamViewer. I've used it with a tablet connected via internet to the laptop/rig setup at home. Works like a champ, but I found the keyboarding on a tablet to be a challenge. Would have preferred a second laptop. The neat thing is your ham software only has to be on the home computer. Any computer with TeamViewer downloaded can be the remote! I know Paula K7PAX has used it for PSK QSOs so she might have some more insight.
As to portable rigs, be sure to consider QRP - something like the FT-817. It doesn't have to be DX to count for 365 and if you can set up a schedule on 40 or 80 with someone in the town you will be in, it almost guarantees a QSO.
I've never tried this, but you might check out the possibility of guest
operation at a club station in the town you will be visiting. The ARRL website has club listings that you can search by city/state/radius that would be a good starting point. I think Steve W3HF has some sort of arrangement with a club in Denver when he is there on business. He's probably a member of their club. Might check with him for insight.
Of course, if you want to dream, or you win the lottery, there is the Elecraft KX2 - built in PSK encode/decode where you send via keying CW from the paddles! 80-10m, fits in your shirt pocket, 13 oz, internal battery, 10w out...optional internal ATU, add an extendable whip antenna and you are good to go...a HandiTalki PSK rig! (If I were rich I'd have one and loan it to you.)
So there are lots of possible solutions. You can do it. Keep us posted on how you make it happen! GL!
Rick - N7WE
070 - #1602