Sorry. The UDL and the Lexers are two separate bits of code, so you cannot just “inherit” or “export” the original lexer to a UDL.
However, you can add extra highlighting to any builtin lexer (like the HTML or XML or SQL lexer) or to a User Defined Language (UDL) using regexes via the script EnhanceAnyLexer.py that @Ekopalypse shares in his github repo … this might allow you to provide the extra syntax that either the lexer alone or the UDL alone won’t give you.