Yep overheating may well be more of a concern than underheating with the log burner. (that and maybe the cat)?
If Carbon Monoxide were a genuine concern then you have more to worry about than keeping chicks warm! We'll assume you have a proper log burner and CO monitor
What sort of room temperature range are you getting through the day and night?
I'd not keep them close to the burner but you'll probably be fine keeping them in the same room dependant on how low the temperature gets at night, they'll soon let you know if they're getting cold
When I was young we kept chicks in the warm kitchen without a heat lamp no problem so shouldn't be a problem.
Chicks will quite happily run around in quite low temperatures outside then pop back to mum for a quick warm up so temperature fluctuation isn't a great problem as long as they can get warmed up again, maybe one end of the basket nearer the burner than the other and move it as necessary?