This works on my Linksys BEFSR41 v3:
Where it says PASSWORD, you'd put your router's password in.
Since we're using UserName:Password in the URL, and the BEFSR41's UserName is blank, you just leave the '' in there. Not sure what to call them... single quotes? ' '.
It'll automatically release the IP address and grab a new one, then redirect you to the Status_Router.htm page of the router, showing you the new IP address.
For some reason, the above doesn't work in Internet Explorer... but it works fine in Chrome and its derivatives.
So you'd create a URL shortcut on your Desktop thusly:
"C:\Program Files (x86)\SRWare Iron\iron.exe" --app=http://'':PASSWORD@192.168.1.1/Gozila.cgi?submit_button=Status_Router&submit_type=release&change_action=gozila_cgi&wan_proto=dhcp --incognito
I'm using SRWare Iron instead of Chrome. SRWare Iron strips out the Google-provided corporate spyware code. I've set it up as an app, so I can include the URL to the router, and I've used Incognito mode to prevent the browser from saving the URL (which has the router password in it).
Strange looking URL, but it actually works!
In regards to the OPs question... I believe you're right... the BEFSR41 has a limited amount of memory, and using BitTorrent on it with too many connections will cause it to act flakey (dropped connections, severely limited upload speed). It exhausts the SPI memory area, so it's forced to drop some connections.
So limiting your BitTorrent client to less than 20 connections, and refreshing your WAN IP address from time to time (which drops all connections and thus flushes the SPI memory) should keep it running good.