Will you be splitting logs, cutting longer branches/logs to length, or merely trimming branches back (where permitted)? An electric (or battery) powered chain saw of saber/reciprocating saw is very effective at many of those tasks. Further, a camp axe simply lacks the weight and leverage (handle length) to do much better than a hatchet when splitting woods. Light duty electrics are close to your price range.
https://www.homedepot.com/p/Remington-Limb-N-Trim-14-in-8-Amp-Electric-Chainsaw-RM1425-Limb-N-Trim/203556884
https://www.harborfreight.com/6-amp-heavy-duty-variable-speed-rotating-handle-reciprocating-saw-62370.html
https://www.lowes.com/pd/PORTER-CABLE-7-5-Amp-Keyless-Variable-Speed-Corded-Reciprocating-Saw/1000061699