If you run the cold water feed line to the water heater to a stripped down water heater tank sitting by the stove first, like an 80 gal, and then the outlet from this tank to the inlet of the heater, you'll have nice hot water out of the heater for quite awhile before running out. And no problems with boiling or anything else because the old heater tank is nothing more than a large wide spot in the pressurized cold water feed line to the real heater. And you don't need a pump etc. but I'd plumb a new T/P valve in the old tank.
Any chlorides in your water will corrode SS.
I did that (in 1973) with an over/under (two on their side) 55 gal metal oil barrel wood burner I built in my basement and a stripped naked 82 gallon gas water heater set up off the floor on 3 bricks. You used to be able to buy a kit for the burner; a door with a draft control for the bottom barrel and a fitting to get the smoke/heat out the top of that barrel into the top one and then another fitting for the stove pipe out the top of the top tank and 4 legs for each barrel.
You may not be able to or want to do that.
In case you'd like to know how I did it, or ideas... I put both in a 3'+w x 6' tall by 6-7' long galvanized sheet metal box with a frame of 1/8" x 3/4" x 3/4" angle. I extended the draft control on the burner out through the door of the box with 3" pipe about 4" long and had a normally closed (in case of a power outage) temperature operated draft control and a another temp controller (high temp safety) in the box to maintain the burner temp. That controller also controlled a fan (on/off) so I could send hot air into my forced hot air heat system, otherwise the heat just rose into the system and it was all control with the upstairs thermostat. I tended the burner 2-3 times a day with dry split oak up to IIRC 40" long. I heated the 2 story 4 bedroom house, with 5 kids, and most of all our hot water use with that for about 8-9 years. I tore out the original oil fired forced air converted coal furnace and used nothing but the wood burner all winter in north central PA. When I shut down the heater at the end of winter, that old heater still tempered the 45* cold water to like 55-70* which saved electricity for the 80 gal electric heater. I used to get 40' trailer loads of logs and would cut split by hand and stack them. I started to go around cutting down some trees for people but gave up on that after a few; tha twas real work.