I've seen quite a few on Android, Modern Combat and NOVA both have a number of games in their series, but I've always found the controls awkward.  Typically what I've seen is a virtual stick on one side for movement, then drag a finger across the screen to look and aim.

The other popular style of mobile FPS is the on rails shooter where you just have to aim and shoot.  Some of these have a stick, some you just tap on the target.

I think your idea of a turn based FPS would be an interesting take.  I love seeing things that are different, so I say it would be worth looking into.

Phones tend to be more casual and play sessions tend to be shorter and more easily interrupted, which makes FPS a less common fit.  There are some FPS games and some games where longer gameplay is possible, but they're less common. As you mention, control is more difficult.

While it is more RPG than FPS, there have been many games with an overhead camera that have proven extremely popular on the platform. Infinity blade was incredibly popular, and these days Fortnite (and to a lesser extent, PUBG) tops of the charts. Lots of combat that plays well on the devices.

Historically the best mobile games are easily approachable and easily interrupted.  The ever-popular Angry Birds is a great example, a single action takes about 5 seconds. Many card games, puzzle games, and trivia games follow the same pattern of being easily interruptible from moment to moment. If you browse through the top ten games at any time over the last decade, nearly all of them have followed the same pattern; they're games you can pick up and set down at a moment's notice. FPS games occasionally make that list, but they're relatively uncommon as they require significant time and attention from the player, in addition to concerns over battery life and device temperature.

There are aspects of FPS and turn based games that make them feel like they should be mutually exclusive. Turn based games, for instance, really require teams of players, other wise the pace is a little too short, and you need the team tactical element to make up for that. Similarly, an fps gives you a limited viewport, which mitigates the range of tactical decisions.

One idea that I thought could work is if you worked integrated the fps role as part of a team. As an example:

Imagine playing the role of a long range sniper. You have, through your scope, a view of the battleground (say an enemy compound). You can see everything, the enemy soldiers, the objective, and your own team positioning themselves at the compound entry points.

The turns alternate between the sniper (you) and every one else (who is AI controlled). In this game, your turn would be looking over the battleground and choosing one target to eliminate. Once your shot has been taken, the game moves forward one step. Your team infiltrates one time unit, and the enemy responds. This would be in realtime, rather than turn based. So you act, everyone else acts.

You could even expand it with a tactical planning phase, where you set waypoints for your team on go-codes (like the original rainbow six games). 

As a gross over simplification. You shoot one guard, your team infiltrates the first room, and encounters a group of enemies, the game pauses. You shoot one enemy, your team opens fire and the enemy reacts, the game pauses... and so on.

You could also incorporate some semi-fixed movement in to the game. So, instead of taking a shot this turn, you opt to move from your current position (rooftop A) to another position (rooftop B).

I think that could work pretty well...