I don't think Dove would be a good choice because I think it might have some sort of moisturizers that might cause problems.
I don't have a front loader either, but when my fleece starts feeling a bit sticky/repelling I wash all my dipes with a squirt of Dawn instead of laundry detergent. I make sure there are no bubbles in the rinse. It works like a charm on fixing repelling fleece on my Fuzzi Bunz and it seems I only have to do it about once a month.
Hmm..I tried this and I had suds forever (rinse after rinse after rinse)--it was horrible and I used a tsp. of dawn. So, try a little if you decide to try it (obviously, it works for some people--I have hard water so I don't know if that was the problem but yikes!)