You can't edit a profile, nor can you do without one.
Generally, the number of colours you need to print is more than the number of colours the printer can achieve with its four or six inks. A dedicated photo printer (with 8 inks) tends to have a larger "colour gamut" (can print more colours). A printer with dye-sub ink tends to have the smallest colour gamut. The profile tells the print software the best match for the colours you're trying to print (if it can match them at all). If the inkset changes, then it won't know which colours would be the best match and the print will look "wrong".
The manufacturer of your dye-sub ink should provide you with a profile for whichever printer(s) they support. It won't be a dedicated 100% accurate profile, but it will be a generic "it's good enough" profile.
If you're technically-minded and up for a challenge then check out Paul's B1100 challenge at this thread:
http://www.dyesubforum.co.uk/forum/view ... f=55&t=268. Before the B1100 was properly supported, Paul set himself the challenge of seeing if he could get it working with an alternative profile (for the D120) - unfortunately the pictures aren't available any more, but they demonstrated quite clearly that the proper profile was needed.
At the end of the day, you have to ask yourself if it's worth all the hassle - or if your time and money wouldn't be better invested in purchasing a properly-supported set-up.
According to Amazon, the printer you're trying to use (SX405) doesn't have a good reputation. Out of 25 reviews, 21 are 3 stars or less (over half are one star). Maybe you should consider switching to the B40W - that one's supported by Sawgrass (Artanium). Then keep your SX405 for non dye-sub printing tasks.
Alternatively, you could try tweaking your current set-up. Using the best profile you have to hand, change settings in the printer driver such as the paper choice. I've noticed on older Epson printers that if you print "Photo" onto "Epson Matte" paper, the print favours green, and if you print "Photo" onto "Epson Glossy" then the print favours red (this is for Artanium ink but it may apply to yours, too). Just tweaking the paper setting might help you achieve a better print but, really, it's not going to work for everything.
I hope this helps a little bit.