Water collecting under shower base tiles

Hi Everyone,
I’m not sure if this should be here or Tiling, but here goes.

Background
12 year old shower built as follows:
· 15mm Hardipanel compressed concrete sheet floor, then
· Puddle flange, then
· Cement screed, then
· Latex waterproofing compound to whole enclosure, then
· Tiles

I did all of the work except the tiling and grouting.

Problem
Recently found water damage to MDF architrave at door approx 1.5m from shower.
Checked under floor and found no evidence of water leak except immediately below damaged architrave. So I assume the waterproofing membrane is good and water is getting under hob tiles and running under floor tiles to doorway.

Rectification
I have removed the 10mm frameless shower screen and removed all silicone from shower.
I have ground out wall and base grout.
The shower hasn’t been used for a week. However, the inner section of the base grout is damp, with water coming through in places (marked in red in picture).

I think the tiler has glued the metal floor grate into the shower floor without leaving any gaps and the water is backing up against the sides of the floor grate.

Questions
Is the water under the tiles a problem?
How do I remove this water? I assume the new grout won’t set while in contact with water.
Should I remove the floor grate and re-glue it, leaving gaps in the glue for the water to drain through? Can I do this without damaging the waterproofing?
Should I drill holes in the side of the floor grate to help the water escape?
Would an epoxy grout prevent the water seeping through into the area below the tiles?

Appreciate any help.