First, I'll comment this has some XY Problem smell to it - as a long time lurker, I'm sure you've read that one. Why do you expect the new token will be less problematic in your process than the existing escape character? How does this factor into a broader parsing problem?

To solve the asked question, split uses a regex to act on a string. Therefore, you should be feeding it a regular expression, not a static character. Assuming you've stored the character literal in $character, you should get your desired result from:

my $stringToProcess = join($SEP_CHAR, split(/\Q$character\E/, $string)
+);
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See quotemeta. If you explicitly only want character escaping in the particular scenario, you can get your desired result with

$character = qr/\Q$character\E/ if $character eq '+';
my $stringToProcess = join($SEP_CHAR, split($character, $string));
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where you store a regular expression as an object; see Regexp Quote Like Operators in perlop.