I'll offer this with very limited experience , dip dye will be the best even result . By all means I would NOT try the whole side , I would break it down to just over the finished size . Use a pair of tweezers to hold it and if it doesn't dye under the point of hold cut off that little bit . Try 1 piece at the finished size , I will bet the dye will still work under the point of hold .
I think I have dipped 3 or 4 items so you may want to wait on someone with more experience