The perm is possible. A really loose demi-wave (wow who's heard those two words together since the 80'S

) would do fine or even just a root perm to add a bit of volume. I would not advise doing that at home - it's way too easy to mess up, especially on bleached hair.
You could play around with curling tongs in the mean time.
Do you have a photo of the cut? It would make it a lot easier to advise if I could see it.
Try this. Separate your hair into a horizontal horse-shoe shaped section from behind one ear to behind the other. Point cut (cut upwards into the hair in small chunks to give an uneven line) the hair into roughly one length. It should look slightly uneven but overall the hair in that section should be about the same length. To check you can use a comb to pull the hair directly above your head to check the length from back to front.
Take the next layer above the last and repeat. Repeat again for the third layer and then either blend the cut at the front by feathering or cut as required e.g. long fringe.
Hope this helps.