To me, it's not the FPS part that bothers me. I'm alright with that.
FPS in x-com should work by (in order of bestestness[sic])
Idea 1 - Taking over one of the characters and also being able to pause, yes, pause in FPS, to command units over a map. If the character you're using dies, you just go to the highest ranked one left. no save scumming, no bringing people back from the dead.
Idea 2 - If you must be a particular character always, at least you should be able to pause and command. If you can't have a map to command them from, think of Fallout 3 VATs. Also, a hardcore mode where if your character dies, gameover.
Idea 3 - If all of that fails and you can't pause, you should at the very least be able to command, give simple orders.
Idea 4 - If you can't do any of that, you're a horrible game studio, give up.
Overall the missions in x-com should be somewhat like this:
Idea 1 - Completely dynamic, the storyline is followed by research results. You can also have some key events or key objects to find in missions, but those should appear randomly. Preference goes to random maps (which only xcomutil modded x-com 1 & 2 have) and very tactical settings that requires team work to win. Ideally, you should be able to send a squad to auto-solve a mission, something all x-com & After series lacked.
Idea 2 - Give room to a few storyline set missions, compromising research and mission to advance the game. Not really fond of this, since replayability is killed somewhat, since you already know the key missions in the next gameplay.
Idea 3 - If you go HL linear story, I will kill you developers.
Idea 4 - Erase idea 3 before people say shit.
As for the base management bit:
Idea 1 - Since it's FPS, being able to walk in the base to spot its defense flaws and be able to assign guards to key locations would be a great thing. But also, keep all base building in a over head mode, same goes to research, nobody wants to fucking walk to whatever to talk to people to get things done. One button beats walking, always. The only except would be being able to move sentry guns and barriers around your base in FPS to create a decent defense line.
Idea 2 - Okay, if you can't have an over head map building, at least try to keep us from walking around.
Idea 3 - No map building? You fail, go to hell.
Idea 4- No base? Just one stupid room with a scientist npc and an engineer npc? Yep, this sounds like this game alright, do not want.