Even if you had a dead short in the RV you still can't go over 15 amps(1800 watts) without blowing the circuit breaker. As Troubledwaters pointed out keeping the peace is also important. An electrician visit to both check your usage and explain how electricity is billed to all of you would be a good neutral ground.
Are you using the house washer, dryer, shower, tub, micro, oven, lights, etc?
You stated that this is their first winter in the house, the Electric company can provide prior usage numbers for comparison.
I would pay half the bill and move on. Also, what does the child of this woman have to say? Sounds to me like you are without backup here.