While not in the hair dryer amperage range, the currents are by no means "small" relative to the size of the center conductor.I don't think "ampacity" comes into play with such small currents.
RG59 features a center conductor that is typically 22AWG and that can handle approximately 0.92 amperes. The 20AWG copper version can do 1.5amperes. RG6, with its 18AWG center conductor has an ampacity of 2.3amps.
The output of the DPP44 Power Inserter is rated at 1.65amps on the output side and that's clearly above the ampacity of either flavor of RG59. It isn't prudent to use wire that doesn't meet the ampere rating of the power supply unless you know for certain what the actual load might be.
The 1000.2 and 1000.4 will work with about 0.7amps.
These numbers obviously apply only to the line going to the antenna.
The node to Hopper cable needs to pass 650MHz to 3GHZ so if the RG59 isn't "special", you may find it isn't optimal. Even today, most RG59 cables are only sweep tested to 1GHz.