How can I ground an appliance with a CEE 7/7 plug plugged into a step-up transformer with a "universal" female outlet?

Question

I'd like to use a European appliance (5A 208-240V 50/60Hz 900W) with a step-up transformer such as this one.

The appliance has a CEE 7/7 Schuko-French hybrid plug. The transformer has a female receptacle seen here, which accepts multiple types of plugs.

It seems to me that a CEE 7/7 plug plugged into this all-female receptacle would not be grounded, since there would be no contact to either the earthing clips or to the female contact for an earthing pin that are on the plug.

Q1: Is it dangerous to use this set-up as is?

Q2: Would it be safer for the transformer to be plugged into a GFCI outlet?

Q3: Could I obtain or make a two-sided male adapter with earthing pins on both sides that would ground the appliance through the transformer?

Q4: Would the appliance be grounded if I cut off the CEE 7/7 plug, and wired on a Type G or Type I plug?

Q5: Could I ground the appliance some other way?

This is in an apartment, so electrical modifications to the source outlets or building wiring are not an option.

Answer 1

After further research prompted by Ken's comment, I see that if I buy either a transformer or a plug adapter with a CEE 7/3 socket instead of the silly "universal" socket, then it will have earthing clips that make a ground connection on the sides of the plug. I missed this before, because I was expecting to see a male earthing pin.

However, I like the form factor and price point of the transformer I originally linked, and would still like to know whether replacing the plug would be as good. Or better, since I had the impression that plug adapters aren't meant for continuous use.