The problem will be in developing the joint strength between the 2x4 and the 2x10. I would put a 4-foot 2x4 edge against the 2x10 with the wide part against the subfloor if there is one. I would use glue with some water resistance and attach the part with three inch "drywall" screws on 4" spacing from the 2x10 side, and to the subfloor with appropriate screw length. You will probably also put one on the bottom.
Actually, the weakening is not as great as it would appear because of some structural arithmetic that I won't cover here.
The hole can cause the 2x10 to split through the hole because of shear stresses. I would reinforce the area around the hole with about a 16" length of 5/8 or 3/4 plywood, full depth, glued and screwed to the side of the 2x10 opposite the side where you put the 2x4's. I would slip the plywood over the pipe when installing the pipe but not fasten it until you get the pipe run, so it will be easier to get the pipe through.
You are going to have a tough time because it will be difficult to get any significant length through the joists. I would design a reinforcement to be installed after the pipe that would let me notch the bottom of the joists and bring the pipe up from the bottom. That would shock the building inspector but it could be made stronger than the original. The result would be a lot like the fabricated joists that you see a lot of these days.
You will need to support the joists before you cut if the joists are supporting heavy loads now. Most floors are designed for 40 lbs per sq ft and may have only about 10 lbs per square ft on them, so you don't have to worry if the loads are light and you are only cutting a couple of joists.
The details of the design would depend on how close the pipe ends up to the upper and lower edges of the joist, and whether you have a ceiling attached to the joists.