Hi,
In case it helps, all of the problems you've listed I've also seen when I've tried to use thinner, not so well shielded HDMI cables in the past. I am not saying that's what your problem is but it might be worth checking the cable to rule it out first if you haven't already. You don't need a super-fancy expensive one, just a decent one that's not super thin or of the flex-ribbon type, as a lot of those are not shielded properly and the signals get messed up as it travels along it. For example, the HDMI cable I'm using at the moment cost about £4-5, is 2m long, and was meant for a raspberry pi. It works fine with the Mk3 at 1080p on my TV, with the output pixel clock running at 148.5MHz.
The display blinking on and off, or not displaying at all could be related to the front/back porch settings or other timings of the output video mode the Mk3 is set to, being slightly out from what the displays expect - What video mode are you running your Mk3 in? Some of my monitors and TVs are terribly picky about resolution and refresh rate settings over HDMI, so yours might be too. If you haven't already, try using the menu option "Settings"->"Re-Read Monitor Display Data..." in the Indivision settings tool, to detect the available screen modes your monitors are advertising when the HDMI cable is plugged in (you might need to have the Amiga connected via the original DB23 analogue video out -> a TV to see what you're doing).
The 1024x768 and 1280x1024 Mk3 presets worked okay for me but caused the screen to get scaled into the middle of my displays and the 1920x1080 60Hz did not work at all for my displays but again that's my displays being picky and not the Indivision's fault.
In case it's useful, these modes work for me...
1920x1080 @ 60Hz
Pixel clock: 148500000
Horizontal Timing:
Visible: 1920
F. Porch: 88
Sync Len: 44
B. Porch: 148
Polarity: Pos
Freq kHz: 67.5000
Vertical Timing:
Visible: 1080
F. Porch: 5
Sync Len: 5
B. Porch: 35
Polarity: Pos
Freq Hz: 60.0000
1920x1080 @ 50Hz
Pixel clock: 148500000
Horizontal Timing:
Visible: 1920
F. Porch: 528
Sync Len: 44
B. Porch: 148
Polarity: Pos
Freq kHz: 56.2500
Vertical Timing:
Visible: 1080
F. Porch: 4
Sync Len: 5
B. Porch: 36
Polarity: Pos
Freq Hz: 50.0000
1280x720 @ 60Hz
Pixel clock: 74250000
Horizontal Timing:
Visible: 1280
F. Porch: 110
Sync Len: 40
B. Porch: 220
Polarity: Pos
Freq kHz: 45.0000
Vertical Timing:
Visible: 720
F. Porch: 5
Sync Len: 5
B. Porch: 20
Polarity: Pos
Freq Hz: 60.0000
1280x720 @ 50Hz
Pixel clock: 74250000
Horizontal Timing:
Visible: 1280
F. Porch: 440
Sync Len: 40
B. Porch: 220
Polarity: Pos
Freq kHz: 37.5000
Vertical Timing:
Visible: 720
F. Porch: 5
Sync Len: 5
B. Porch: 20
Polarity: Pos
Freq Hz: 50.0000
Last thought, (which I should have asked at the top of the post)... have checked you are updated to the latest version of the firmware and tool?