You'd probably not be allowed to use your gas cooker. The following answer is written regarding the Stockholm County (as the counties are responsible for the bans). But as the bans are based on the recommendations from the Civil Contingencies Agency, they are most likely similar in other counties.
The information can be found here and states that:
Förbudet omfattar alla former av öppen eld, till exempel att grilla på iordning- ställda grillplatser samt att använda trangiakök, stormkök eller medtagen grill.
All types of open fire are prohibited, eg BBQ-ing and using “outdoor kitchens” or your own BBQ.
The FAQ on Stockholm County (Swedish Only) states that:
Får jag använda friluftskök i tält? Ja, om tältet står på din egen tomt. Annars inte eftersom tält står utomhus på marken.
Can I use an outdoor kitchen in a tent? Yes, if the tent is in your own yard. Otherwise not, as the tent is outdoors, on the ground.
However, the same page also states that you might use an outdoor kitchen on a camping ground, provided that the kitchen is on a platform and not touching the ground.