While the Fake Sneak Placement mode is enabled, I'm completely unable to place any Item Frames. Right clicking doesn't even play placing animation, and it works neither on regular blocks, nor blocks with GUI. Even holding shift doesn't work.

This seems to be the case on both singleplayer and multiplayer.

This is a consequence of how the Fake Sneak Placement feature works - it offsets the click position to the air block infront of the clicked block, to avoid doing any possible use action on that block, such as tuning a noteblock or opening an inventory GUI. But the item frame and painting for example need you to click on a block, they won't work when clicked on an air block.

I guess I could make the Fake Sneak Placement only offset the position if you are holding a BlockItem. Especially now in 1.13+ after the flattening and the related code improvements/clean-up, such as all blocks now having an associated BlockItem instead of "special regular items" such as redstone dust and I think also repeaters etc. in 1.12 and before.