Maybe construct a plywood box and stick beading or split tree clippings on to create the log effect.
Much simpler. And no toxins. Glad I thought of it.
Actually, the issue is making the corners appear as if the logs are interlinked. That is, like the old Lincoln Logs looked after putting them together.
Still, I find your suggestion, Cassandra, superior to paper mache.
I've threatened to build a birdhouse for use at my DD's log cabin. Though a miniature replica of her abode is out of the question, I still fancy making the birdhouse resemble a log cabin. My internet "diy log cabin birdhouse" search results show quite a few options from products to kits to books to videos.
But nothing that seems to be within my woodworking skill set.
One alternative I'm considering is, instead of doweling or local tree clippings, building the birdhouse out of paper mache. But the recipes I find to make paper mache waterproof involve chemicals that I fear may be toxic to small avians. One potential solution is to create a plywood box as the interior and fashion the waterproofed paper mache over the box.