You need a river or lake, even a pond will do for a small critter. You will need to create a raft or other craft that you can pile with combustable woods like kindling - fatwood comes to mind, or pine. The craft should have a sail mounted forward of the fire wood so the wind can take the craft and critter as they are both consumed by fire. You could even create a raft type thing from fatwood so the entire thing is consumed.
It will also be necessary to prepare the body so it does not float back to the surface, as in India in the Ganges, after the craft goes under - this is rarely spoken to in Norse mythology as most Kings who were treated to this process had already suffered fatal sword/stab wounds all over but specifically to the abdomen in combat, and/or had their entrails read by the tribal medicine man, or whoever, after being sacrificed in hopes of enhancing crop harvests.
Also, most of those men were decked out, so to speak, in all their battle gear which weighed a lot and would probably drag them right to the bottom without the benefit of abdominal eviseration. You could always, ummmm, 'secure' your soon to be departed beloved pet to something less bouyant that you pile the wood around so that as the fire consumes both craft and critter the less bouyant object slides quickly to the bottom..............
Now understand, please, that this didn't work for us in Vietnam when members of the 5th SF Group sank a double agent into Haipong (??) harbor but that's only on accounta the dang seals went in and recovered partial remains, unrecognizable except for a scar on the guy's toe according to the widow (right), which led more to embarrassment than convictions in court martials. After that we just fed'm to the hogs.........
So this process should work for you unless there are grieving family members who would want to make certain the rat passed of natural causes and was not 'helped along' before his time...................... HTH.