After being confused as to why there isn't a dedicated Flare button for controllers despite there being one button always unused on Tank and Modern Controls (I hope that is patched at some point regardless of the below shortcut) I accidentally figured out the button combo for getting out a flare, and through more button combos, shortcuts for weapons too.

To use the shortcuts you have to hold down the button you use to open your inventory and then tap the following button to get out a Flare or equip the weapon you want. If you use the Flare Shortcut again whilst Lara already has one in her hand she will throw the one she currently has in front of her, useful for throwing Flares down dark shafts to see what's at the bottom of them. 

Tomb Raider 2/3 Flare Shortcut:

PlayStation: R1
Xbox: RB
Switch: R

Pistols:
PlayStation - X
Xbox - A
Switch - B

Shotgun:
PlayStation - Circle
Xbox - B
Switch - A

Magnums:
PlayStation - Square
Xbox - X
Switch - Y

Uzis:
PlayStation - Triangle
Xbox - Y
Switch - X

Pistols:
PlayStation - X
Xbox - A
Switch - B

Shotgun:
PlayStation - Circle
Xbox - B
Switch - A

Automatic Pistols:
PlayStation - Square
Xbox - X
Switch - Y

Uzis:
PlayStation - Triangle
Xbox - Y
Switch - X

M16:
PlayStation - L1
Xbox - LB
Switch - L

Grenade Launcher:
PlayStation - L2
Xbox - LT
Switch - ZL

Pistols:
PlayStation - X
Xbox - A
Switch - B

Shotgun:
PlayStation - Circle
Xbox - B
Switch - A

Desert Eagle:
PlayStation - Square
Xbox - X
Switch - Y

Uzis:
PlayStation - Triangle
Xbox - Y
Switch - X

MP5:
PlayStation - L1
Xbox - LB
Switch - L

Grenade Launcher:
PlayStation - L2
Xbox - LT
Switch - ZL

Harpoon Gun (Equips automatically when weapons drawn underwater with Modern Control scheme but doesn't with Tank Controls):
PlayStation - R2
Xbox - RT
Switch - ZR

Rocket Launcher:
No Shortcut. If Flare is given its own button in a patch this could be assigned to R1/RB/R?

Great to see these PC hotkeys from TR 4 adapted and brought over for controller users into these Remasters. I imagine PC users should be able to use numbers 1 - 8 as Hotkeys for their weapon equips like in TR 4 but I haven't tested it - currently playing on Switch and will play on Steam after. Hopefully they also adapt the Small Medipack and Large Medipack hotkeys into controller shortcuts from TR 4. Those analogue stick clicks are unused so, why not?

Happy raiding!