I too recommend lipo batteries. Just make sure any buck regulator is up to the job you ask of it, testing is essential as specs can be aspirational.
And be sure to test at the low point of lipo voltage sag. It may be necessary to use a 3s pack, 2s might go quite near 6 volts under load at the end of charge.
Use real batteries and real loads for tests. At least real batteries even if just resistive test loads.
Buy good regulators and you get closer to specified performance. Again, trust but verify.
You can probably cut you budget figures down by promise in software and design to not run every servo on board at its stall current requirement at once.
a7