I don't know the code requirements regarding voltage applied on the wire type installed etc in your area however there is a practical solution to dramatically improve the the amount of available power and reduce voltage drop.
If you can connect the existing feeder to 240V (two pole breaker instead of one pole breaker and neutral) and install a 240:120/240V transformer (like this https://canadatransformers.com/5-kva-isolation-transformer-sc5c-k.html) to supply your panel you will cut the amperage on the feeder in half. The transformer would make your shop panel a new source and require the transformer neutral to be grounded at the shop panel as there is no longer a neutral connected back to the house. Others familiar with US electrical code would need to add their 2 cents regarding legality but electrically this is fine assuming the feeder cable is suitable for 240V. You would also have 4800W available instead of 2400W that you currently have with the 120V supply.
At a minimum, you would need to ensure any other breakers between the shop and main panel are also 2 pole and there are no other 120V loads fed from this circuit between the main panel and the shop.
Power factor correction capacitors will reduce the current required to run that motor slightly but likely won't be a noticeable improvement and it does nothing for inrush on an AC system.