Everything you suggest, can be done. It is a bit impractical though, the best is to start from the right raw materials.
Soda glass can be purified by high temperature. The sodium oxide will evaporate as the temperature becomes high, but we are talking on the upside of 1600 $^o$C perhaps even up to 2000 $^o$C. All the other alkali and many rare earth oxides will evaporate as well, so if you want any of those back, you'll have to add them. In addition, high temperature silica is a mess to work with when it is pure, viscous and prone to "boiling over" when releasing gases - so YMMV. So there you have pure silica. It'll never be "fused" silica though, the name derives from the process it was taken from - and it is usually in the form of micro-sized globes or spheres. You'll end up with a "lump" or "pile" of silica.
To make lead glass, you have forgotten an important ingredient - lead oxide. So, solidify your silica glass and crush it. Take a batch of molten lead oxide dissolve silica the in it (there is something with "molten lead oxide" that is depressing and discouraging but don't let that keep you from your plan). One should respect lead and its oxide, especially when there is an appreciable vapour pressure of them - you'll be permanently heavy metal poisoned if exposed. So, as the viscosity of this mix will become higher as you add silica in it, you are going to have to apply generous amounts of elbow grease to get this mix. I take no responsibility for your elbows though, take care when sloshing around hot and sticky liquid-ish substances.
Now, with all that being said - use the right raw materials and it is much easier. Still not easy, that's why money can be made making glass - but easier.