I agree that trying to do this with 12v batts is already breaking the bank. 2KWh out of cold lead-acid batts is extremely expensive!
You can get a propane RV furnace which is a heat exchanger, which means it separates combustion air from the inside air. It requires 12v for the fan. They have automatic electronic gas valves and starters, too. They use a thermostat input- an unpowered mechanical thermostat connects 2 wires when a bimetallic spring cools down and bends. Heater turns on gas, waits, ignites, waits for thermocouple to show flame. No flame, retry. Repeated retries, no flame, kills the gas to keep from blowing something up. Just like any other home furnace.
But you don't necessarily NEED a heat exchanger. The catalytic radiant heaters are much cheaper, although they lack the automatic electronics of course.
Plants won't mind the exhaust, as long as the combustion is complete. In fact they love it! CO2 is plant food!
However, if the space to too well enclosed and the heater runs for a long time, it may use up so much oxygen that the mixture becomes too rich and fails to burn completely. In this case CO could be formed. Plants actually don't mind CO, but it'll poison humans if the level's high enough. It wouldn't take a lot of ventilation to keep this from happening, though.
Honestly? I don't know about the UK, but in the USA, natural gas heat is MUCH cheaper than electricity. You might forget the electric heater and just put in a small gas heater and turn it on for the entire winter. And again, plants thrive on elevated CO2 levels.