You found the right registry key. On my system it looks like this:
Windows Registry Editor Version 5.00
@="\"C:\\Program Files\\Mozilla Firefox\\firefox.exe\" -requestPending -osint -url \"%1\""
If you change the line at the end to read -requestPending -osint -profile \"full path to your profile here\" -url \"%1\""
it should work correctly, once you save this as a .reg file and merge it in.
If you want to make the edit yourself using regedit, simply insert -profile "full path to your profile here" between the -osint and -url parameters.
Note that in this case, the quote marks do not have preceding backslashes.
EDIT: I just realized there are two occurrences of the FirefoxHTML key, one under HKEY_CURRENT_USER\Software\Classes and one under HKEY_CLASSES_ROOT. You can choose to edit both simultaneously, or try just one and see if it's the one that sticks.