Among other solvents that might prove possible, methanol is stated to have some solubility for NH4NO3 and Ca(NO3)2. (Ref: CRC Handbook) However, there is a very good reason to NOT use methanol with soluble metallic nitrates, especially when the nitrates are acidic (like NH4NO3). The mixture would be highly flammable and very likely susceptible to self-ignition if you stirred it too hard or heated it near any source of sparks or flame. The safety hazards are overwhelming. Any other replacement solvent that is flammable or contains carbon (even if not very flammable) could be a serious hazard.
However, there is a way to use water and get a fairly pure solution of ammonium nitrate. You can calculate the exact amount of NH4NO3 needed to react with a given amount of Ca(NO3)2. You will, of course, need to know the exact concentration of your ammonium sulfate solution and the exact amount of water in the Ca(NO3)2 (because it is generally supplied as a tetra hydrate, but that varies, depending on whether you use a research grade or a technical (fertilizer) grade).
Now the problem is that poorly soluble calcium salts are generally more soluble in solutions of ammonium salts, and here we have one of the reactants and one of the products being an ammonium salt - you can't get away from it! Or can you? The reason Ca(SO4)2 is (somewhat) soluble in ammonium salts is that the acidity of the ammonium salt converts some of the SO4$^{2-}$ to bisulfate and some more of the calcium sulfate can dissolve. If you add NH4OH to reduce the acidity, you can eliminate the solvent power of the ammonium salt. In this way, you can complete the precipitation of the calcium sulfate.
There will be a price to pay: now the solution is going to smell like ammonia. However, if you do the addition of ammonia carefully, to about pH 7-8, the odor should be manageable. Use good ventilation. Filter off your calcium sulfate precipitate and evaporate your ammonium nitrate solution with minimal heat but plenty of ventilation.
Further purification of the calcium sulfate could be done by simple washing; the ammonium nitrate could be redissolved, filtered and evaporated to produce crystals as opposed to evaporation to dryness.