Suggestions for a strong but pet friendly weed killer please.

What is a flame thrower?
It sounds dangerous in this dry weather.

If you're going to use J52's method (which is great on paving) I'd miss out the salt as it will affect the soil health and replacement plants may struggle.

Alan Titchmarsh advices to hoe up seedlings as they appear and leave the roots to die. This works especially well in hot sun with annual weeds. Perennials with deep roots are another matter. Euphorbia is a menace.

Can you cover the area with layers of thick cardboard?

It will prevent regrow the and then you could use the flame thrower later in the year if necessary.

Covering with cardboard seems to be working well with the marestail and bindweed on my allotment.

A vinegar and salt mixture might work. Rough up the soil first and then sprinkle the salt ( dishwasher salt is cheap ) then douse the area in vinegar. The small quickly goes. Leave to soak in for a couple of days and then repeat.
It works on weeds between my paving so should work after a few douses. Completely safe, non toxic, although I wouldn’t let my pets on it until it is dry.