Does it matter who I send to repair the tower on Palaven's moon?

It makes no difference. It just comes down to who you want by your side while you fight off the wave of husks, an event that lasts only a minute or two. After that, your party member will rejoin you.

If you could lose a party member this way, I would have happily used it to kill off James Vega.

Surprisingly, yes. The more technical minded Liara will complete the objective faster then the meathead James. This will result in one extra round of husks because James is far slower then Liara about fixing things. He'll succeed of course but you and Liara will face extra husks for your troubles. But no, you won't lose anybody.

This happens later on in other missions but if you're tech heavy (such as having both Tali and EDI in the squad) it doesn't matter.