At elementary level, you don't really need a native speaker. What you do need is someone with a good pronunciation. That means: avoid Canadians and French from the south of France, as they tend to have very different accents. A non-native speaker could at times be more effective, as he/she will use a simpler vocabulary. Once you reach an advance level of French, you'll be able to profit from a native speaker.
French grammar is not so difficult. For an English speaker, your main problems will be the concordance of the article, adjectives and verb with the subject, i.e. distinguishing between male/female, and singular/plural.
I grew up in Belgium, a profoundly bilingual country (Dutch/French). I'm Dutch speaking, but worked many years in French, and was married to a French speaking Belgian. I have been a teacher for nearly 20 years (albeit not a language teacher).
I doubt whether you need extra classes. I would imagine you rather need coaching. Thus, leading the instructor where you want him to go, isn't as bad as you think. In fact, it's all about you and your particular needs. By all means, start with having your pronunciation corrected. A perfect sentence with the accents wrong, will be incomprehensible for a French speaker. Having a perfect 'sound' will prove to be much more important than using a wrong gender. Your biggest problem will be to find an appropriate time slot between the US and Europe or Asia. I'm at GMT+7, which could well be opposite of where you are.