You could do something like this, but I don't think a 70 amp-hour (Ah) battery will be up to the task – assuming you only want to charge it once a week. As a general rule you don't want to discharge the battery beyond 50% (35 Ah in this case) and capacity goes down as the discharge rate goes up. The capacity rating you will need will also depend on the type of battery that you're using (a traditional flooded cell lead acid battery would need to be larger than an AGM or gel battery). So the first thing you'd want to do is to up your battery capacity, around 140 Ah for an AGM or gel battery or around 220 for a flooded cell. You can use this calculator to experiment with your options.
The advice to go with a DC heater is good, heating loads are generally DC anyway, the only practical difference between a DC heater and an AC heater is likely to be the fan motor and the resistance of the heating element (to account for the higher voltage). Staying DC will both save you losses in the inverter and money that can go into a bigger battery.
To do this project safely you'll need to find a secure way to mount your battery, and you'll also need to plan for both over-current and over-temperature protection (the heater(s) may already include the latter).
I'd be inclined to design the remote heater control around a small capacity relay that operates a larger relay for the heaters. I think that would help hold the cost down.
I'd also suggest looking into using the heater to heat the water in the heater loop of the engine cooling system and using your actual heater/defroster to handle the job. I think you'd be less likely to have fire problems and you'd save space and mounting issues. It would also help to warm the engine so that you'd have heat once you start driving.
All of that said, if you have AC power available, I'd suggest looking into using a block heater to warm the water in the cooling system. That would give you heat right at the start and would have the engine already up to (or close to) operating temperature. If you wanted to defrost you could add a relay to bypass the heater fan switch and run the defroster as you are planning to do.