Synonym For End of the Line, Last Person to Call
Solution 1:
How about your last-resort plumber?
From the OED:
last-resort, adj: Used or adopted when there is no other option or when everything else has failed.
It can also function as a noun:
last resort, n: A final opportunity for appeal. In later use more generally: a final expedient, something to be adopted when everything else has failed.
It does imply that it is the least-desired option (otherwise, why would you not choose this option earlier?), but does not necessarily imply that you can't "count on" it. It certainly captures "you don't call on them unless it's dire".
Solution 2:
Fallback [fawl-bak] /noun
- something or someone to turn or return to, especially for help or as an alternative
/adjective, Also fall-back
- of or designating something kept in reserve or as an alternative
Source: Dictionary.com
Solution 3:
"Last-resort", as previously mentioned is a good answer.
Other options are:
"Last-chance" plumber.
or
"Emergency" plumber.
or
"Go-to" plumber. (Although this implies that you would call him first. It sounds like the plumber you're talking about is not a very good plumber and that is why he always answers the phone, and he's your last choice.)