The 722 would be able to power the switch. Power inserter would not be needed in your case.
The 722 will NOT power a DPP44 switch properly. Especially if he has some variant of a 1000 dish. The DPP44 switch is expecting 28VDC @ up to 2Amps (Up to 56 Watts). Most of the time, this is overkill for the 44 switch, especially when used with only 2 LNBs. The ViP722 receiver has a Maximum output of about 19VDC @ 750mAmps (Up to 14 Watts). That's only 1/4th of the expected power available. That's probably enough on it's own to power the three LNBs for a 1000 dish (or 1000.X) but the switch also uses a fairly significant amount of power for band translation, band stacking and amplification. Will it work, sure. Will it work for a long time? Sometimes. Is that the way it's supposed to work? No.
A good Analogy is the people who put large audio amps in their small vehicles with small Alternators (the ones who's lights dim every time the bass hits) It works, and it works fine for possibly a long time, but it puts more stress on the alternator, and the Amps, then it would if there was a proper amount of power available. This will lead to premature death for both components. In the same way, powering the 44 switch with the 722 will put extra stress on the 722 power supply, and the 44 switch, possibly leading to them dying sooner. Also, if the 44 switch does not have enough power available for the LNBs, they will be fed less power than they are expecting possibly causing lower signal strength, or premature LNB death as well.
I do beleive if you talk to the right person at dish (tech support, not a CSR) They will RA you a power inserter. By all means, enjoy your dish with the 722 powering the switch until you get a replacement, but install it right as soon as you can.