AFAIK the volume size shouldn't effect the backup time that much. Actually in some cases, smaller volume size can increase the upload speed (very much depends on the quality of your internet connection).

Depending on your scenario, a larger volume size, may make a marginal inprovement, but it's most likely your real world upload bandwidth to AWS S3 that will be the bottleneck. That will depend on your ISP and how much traffic S3 is getting from your region. Just because you have a "fat pipe" won't necessarily mean you'll get superfast upload.

TBH I'm not sure how many threads TKLBAM is capable of using, but to saturate your bandwidth, more threads = faster. Obviously that will depend on how many threads your CPU is capable of handling (generally how many cores it has). Keep in mind though, if you saturate your upload, most other things will grind to a halt (e.g. web browsing etc).

I'm not sure what is in your backup, but 160GB is massive! FWIW most of mine are less than a GB (compressed). Having said that they are all fairly low traffic sites without a ton of customisations. Regardless, perhaps it's worth trying to tune your backup a bit?

An alternate (or additional) approach is to split your backups and have multiple smaller ones. If you go that path, I'd be inclined to have a "master" one that includes all the server config and the "main" app you are serving. Then another which contains all the massive files (that hopefully don't change that often?). It may take a bit of trial and error (and make sure you test everything thoroughly!) to get something that works how you want...