Competing with an amateur partner is not that expensive until you get to the highest levels.

Provided that you focus on what you really need to know for beginner to intermediate competitions and don't get distracted by overcomplicated routines, one private lesson a week is enough - especially if both you and your partner can contribute the full cost of a lesson, giving the partnership two per week (couples lessons are the same price as singles, or in a few cases slightly cheaper, since the teacher doesn't have to do as much physical work).

Most beginner to intermediate couples competitions do not permit fancy costumes anyway (in contrast with pro/am competitions which usually permit and expect them at all but the most entry level).

Unlike with pro/am you will not be paying for a teacher's time at the competition, which is a huge savings. Entry fees tend to be $50-100 per person per competition, plus around $60 to register as a dance competitor each year.

If you want to do this, the most important thing besides finding a partner is finding a studio/teacher who already has a few amateur couples - otherwise you are likely to spend a lot of money on help with the wrong things, and turn up to the competition to find out you are in violation of one rule or another.