Can't easily flash your TSOP with the CheapMod without additional code and a 12C508 addon.
Your best bet is to use the 007 trick and Raincoat.
Since your Xbox is already modded, you won't need a memory card. Just copy the appropriate save file to your Xbox hard drive via FTP. Bridge the TSOP write enable points, boot 007, load the savegame from HDD, and flash. Better yet, split your TSOP first using one of the excellent tutorials.
But why flash your TSOP if you already have an LPC mod? And possibly sacrifice your ability to play on Live? For most of us here, multiple BIOSes are unnecessary. If you require multiple BIOSes, buy a cheap used X2.0 Lite for under $20.