Pardon the crude Grid Frame...

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xx Grid Frame, 1st Name (alphabetical) xxxxx
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xx Grid Frame, 2nd Name (alphabetical) xxxxx
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I'm not positive if this is the exact method to set this up.
I read a post of this recently, but have not implemented it into my play.

You create a pair of mapped keys.
Each is two steps, executing a step on press or release.
Set one of them to press and hold Alt on step 1.
The other for Ctrl, on step 1.
Both of them aren't pressing anything on step 2.

Create two repeater regions.
You have the option of setting a mapped key to execute when the mouse is in the region.
Pick the one of the two (Alt/Ctrl) for the left region.
Pick the other for the right region.

When your mouse is in the left half of the grid frame, any click (left/right/middle/scrollwheel, etc) is the same as doing it with Alt also held.
Ditto for the right half, except as if Ctrl is held.

Then in Clique you set your click interpretation to be based off of Alt + Clicks, Ctrl + Clicks, etc.

In play, move the mouse over the left or right half of the Grid frame.
And click the relevant mouse button, no need for any keyboard input.
And it will count as Alt (or Ctrl) + Whatever click.