I have a sleeping bag that is 6 degrees Celsius- would that, with some warm blankets, be warm enough for the burn?
Should be fine, with the blankets. It does get down to around 0c some nights and I was chilly even in my -1C sleeping bag. Last year was particularly warm so it was unneeded, but who knows what the year will bring.
For a mattress ... I've literally used the $8 Walmart Intex mattress for like 30 camping trips since 2013. It works fine and is quite comfy. It WILL stretch and relax when you first use it (for the first few days), and it will stretch out if it's in the sun, near the edge of your tent, so you will have to top it up each night... but not really a big deal. Don't expect to fill it once and have it last all week, but a handful of pumps shouldn't really be an issue.
It's a little bit more expensive on Amazon but still only $15