You may at first notice the heat bothers the swelling a little bit but you will be placing your feet in ice cold water too. So I wouldn't worry about it. You must use hot water to get the circulation going and get that fluid moved out.
Peppermint oil needs to be full strength. Don't dilute. It may make you feel a little cold until you get use to it and you become less acidic.
If you notice too big of a reaction of the peppermint oil with the acids on your skin, then I would try wintergreen oil. It is much milder, with a smaller amount of menthol in it....then later on you could try the peppermint oil again.