I did a fairly extensive Google search and found only two examples of the construction with the full infinitive. It is safe to say that if one wants a causative meaning, the only recognized form of the construction is
HAVE + (noun or pronoun) + bare infinitive
Therefore the only correct version is
We should have a plumber see to it
Now, if one inserts an adverb of place after the direct object the full infinitive is possible, but it changes the meaning:
We should have a plumber HERE to see to it
Similarly, an adverb of time immediately following the direct object changes the meaning of the infinitive:
We should have a plumber TOMORROW to see to it
...but the meaning of the infinitive "to see to it" in both cases is "for the purpose of seeing to it." It does not imply that the action of the infinitive was actually carried out. This fact becomes clearer when we use the past tense:
We had a plumber yesterday to fix the shower, but it was beyond repair. We have to get a new one
In these cases, there is no grammatical connection between the main verb HAVE and the following infinitive. It is not a causative construction.
The causative construction with the full infinitive may occur in some regional dialects, but it is not standard English. Both Quirk et al.* and Biber et al.** state that the only form of the causative construction is with the bare infinitive.
* Quirk et al., A Comprehensive Grammar of the English Language (Longman, 1985)
** Biber et al., Longman Grammar of Spoken and Written English (Longman, 1999)