Placing a mounting hole partially outside the board

That's why you should always place the mounting holes before you place your components :)

The good news is that the manufacturer will not have any issues making these holes. The only thing I'd suggest is to remove the plating.

Another way to do this is to define the "mounting slots" on the PCB outline instead of drilling. That way, you'd have greater control over the shape of the holes.

I would suggest talking to your board vendor. What you propose can certainly be made but different vendors may have different preffered ways of doing it.

Drilling it as a plated hole and then cutting would likely be the cheap and dirty but risks the plating getting ripped off and leaving an ugly mess.

Drilling non-plated holes and then cutting means an extra drilling step which some cheap board vendors may not like.

Drilling may also require bigger gaps to be left between the boards reducing the efficienty of panelisation.

Making it part of the board outline would be a good option if the board vendor intends to mill the board. Some vendors use milling as their default de-panelisation method, for others it may be an extra.

You could ask yourself whether you need mounting holes that big? Apart from the top left corner it looks like you could bring a smaller hole further into the board?