Would it be safe to try using a detergent within baths to soften up the skin and possibly dislodge the scabies mites?

Just use epsom salts they work great for post scabies! They are about 3-4$ for a 2kg jar and each jar will give you about 4-5 baths at one cup per bath. I recommend staying in for atleast 30-45 minutes and at night time, also make the water hot or warm. This will help your skin heal and allow you to get some rest at night!

I am pretty sure a high percentage of people are allergic to overuse of laundry detergent just in the washing machine, so bathing with laundry detergent will not do anything for scabies or your skin, it will likely make it red and allergic even more. Try using Johnsons foot bath soap, baking soda, or my favourite epsom salts. Those will help you skin in post scabies and open up the pores to help surface out the debris. I would also get a sulfur soap to use in the shower and then use cocoa butter lightly on places without lessions, I say lightly because if you use too much you will clog your pores and end up with more eruptive type skin.

Detergent is made to put in your washer to wash your clothing not to put in a bath. Likely it will just irritate your skin and make things worse or even allow the scabies to thrive more on your skin and duplicate. If your trying to ease the scabies conditions such as itchiness you should be using epsom salts as they contain sulfates and are great for healing your skin!