I don't understand why this error is even being generated!
A string is defined by the user/programmer, not the compiler. It should be up to the user/programmer to decide what goes into a string, and the programmer must parse strings accordingly.
I have ASCII 0x07 in my strings, and I need it there!
How can I get around this? Do I need to search/replace all of them and escape them?
Yes, you will need to escape them—as you say, it's a simple find-and-replace. Code is read more often than it's written, and by construction, readers of code can't see what's invisible. Some have even asked about extending this restriction to include Unicode characters in order to mitigate CVE-2021-42574 ("Trojan Source" attacks).