Strzelecki or Birdsville Track?

What about taking the Birdsville, through Mungerannie (fuel up), then the Warburton and through the Simpson on the K1 etc tracks and on to Birdsville. Then you can come back down the Birdsville Track and turn left through Walkers Crossing to Moomba area and Innamincke to see the Dig tree etc area and then swing east.

Do it all. But that would take time and you didn't tell us if time is an issue.

I also don't know how much fuel you carry. I hold over 200lt in two tanks so this would be easy for me.

PGU, do you have a Desert Parks Pass ?
Pretty sure you will need one to do the K1 to Poeppels drive (or anywhere in the Simpson Desert Cons Pk in SA).
You will also need an approved sand flag setup as per Nat Parks regs.

If so, all good, you won't risk a fine.
If not, and you don't want to outlay $150 for one, then you could do Inside Tk and either back down BV Tk / Walkers Crossing as Phil suggested, or go around Cordillo Rd to Inna etc.

PGU,

If you are travelling through the Gammon Ranges, you could continue on passed the Arkaroola turnoff in a NNE direction and come out onto the Strzelecki Track near Mount Hopeless. From there it is about 20km to Montecollina Bore where you can camp, or continue on to Innaminck.
Innamincka to Birdsville via whatever track you want to take. I prefer the Cordillo Downs Road because I think it is a more pleasant drive. I haven't been into the Simpson Desert so I can't help with that.

Whatever way you go will be enjoyable.

Regards,
Bob

As always just a bit of rain can close any outback track so check road conditions before you press the launch button.

As an aside, Balcanoona to Mt Hopeless on the E side of the northern Flinders R doesn't get much traffic and should be an easy drive when dry.