Do use Mosque..don't take it in other way...but some time, i also face same situation with my kids...so i look for near by mosque. there i use to request some buddy to use there toilets for my child...

i thing u will find mosque in every 1 km in qatar....
so thats why may be they don't built public toilets..as they are already available in mosque..this is my point of view..

with all respect to Muslim Culture & Qatari Culture..i don't know the outsider's can use the mosque toilets or not. or any non-muslim can use it or not. but i will prefer to ask somebuddy, whenever my kids emergency happen...

correct me if i am wrong.