Norman,
From a loss standpoint, almost anything will be fine over the 10' length you
mentioned. The real issue is the quality of shielding, and the placement of the leads. You will be operating in the near field of the antennas, and you want to minimize RX pickup from the TX antennas before you get to the preamps. RG-213 is a single shield coax, while RG-214, Bury-Flex, and RG-9913F have double shielding. While the double shielding is better, especially if the cables run close and parallel to each other, many people do fine with RG-213. Note that the generic versions may have significantly poorer shielding.
The UEK-3000 has 30db gain, and an 80' run with the 2 meter IF output should not be a problem. Back in Ye Olde AO-40 Days, I had a 70' run of RG-8 down to the shack. It was there, I was in a hurry, and it worked. (If you have one with the 70 cm IF, the answer is less certain.) Be aware that RG-6 is 75 ohm rather than 50 ohm cable, the big brother of RG-59, so the SWR on the line will be a bit higher. However, that should not be a problem.
Glad to hear someone else is keeping the faith on HEO.
Alan WA4SCA