Is it ethical to invite a student to apply for a job through my spouse?
(Disclosure: I'm not a professor.)

I don't personally see any ethical concerns SO LONG AS you indicate clearly (as you said you would do) that the recommendation letter will be equally strong regardless of whether or not the student applies through your spouse. Under those circumstances, I'd call this "networking," not "bribery" or "coercion". 

The idea you proposed of writing the letter first and then informing the student seems like a nice way of ensuring that the student knows you've written a strong letter. There is the possibility that the student will still feel pressured, of course -- "if I don't say yes, then my professor will send a second contradictory letter" -- but assuming you have a positive relationship with the student, that seems unlikely.