I don't know the best solution off hand, so until someone else suggests something they know for certain will work, I'll give you some things to consider.
Calcium oxalate is listed as soluble in dilute HCl and dilute HNO3, but you've tried the former and found it not to your liking. Oxalate forms soluble complexes with iron (III) at pH < 3, perhaps producing a soluble mixture of dissolved iron oxalate complexes and dissolved calcium chloride, so adding a little bit of Fe(Cl)3 to your HCl may help- if your piping will stand its severely corrosive effect!
You've tried HCl and EDTA, but what about both at once- the former to get the calcium oxalate into solution, the latter to strongly chelate the calcium and keep the oxalate precip from re-forming? Perhaps this will work too, without the nasty iron chloride.
I'd stay away from sulphuric acid because you may be trading your calcium oxalate scale for calcium sulphate scale.