I'd be happy to help make this more efficient though (at no charge of course). Offhand I'd say "web server which accumulates data and uploads it to S3 every N requests or M seconds" would probably get you what you need at a tiny fraction of the cost of "lambda which posts to SQS". Create an AMI and toss it at an autoscaling group and you really won't need to worry about scaling issues either.
For us, the cost works and we have appropriate margin for it. The cost savings aren't worth the extra "we have to monitor these servers" thoughts. Our approach is 100% emotional.
The offer is open if you ever change your mind (e.g. if the cost becomes annoyingly large when you scale up further).
You might want to take them up on the offer, you don't get an opportunity like that everyday.
Absolute legend.
But seriously, it's clear making this more efficient isn't a priority for them, and I completely understand and respect that. I ignore plenty of good advice for the same reason: "Working on more important stuff in Tarsnap right now".
Fathom has a $34/month plan which handles up to 400k pageviews per month. 400k Lambda invocations is 16 cents, if they're doing it efficiently. Transfer pricing seems negligible also at only ~2KB a request.
I'm pleasantly surprised, actually, but of course when someone throws millions of improper requests at you every second.. I guess that's when the game changes ;-)
But if you're regularly rebooting servers because they stop responding, something has gone very wrong.