There's many parents who have learnt Esperanto specifically to teach it to their kids so that their kids are raised bilingual, actually. I'm not sure how much info you can find in English on it, but there are YouTube videos and websites for this among other things (also, I'm not sure if such sites are entirely in Esperanto or not, but most likely they don't have English...). I specifically remember a Dane who was asking in English on Youtube, what people thought about teaching Esperanto to his kid (and then he did end up making his kid a native Esperanto speaker), but there should be more out there and it would be fairly easy to just contact them and ask if they know about studies. There's even conventions or sections of conventions intended for parents with Esperanto-speaking children, so in the worst-case scenario you can just visit one of those and ask people there (as they themselves might have even been involved in research projects). You can also try writing to any English-speaking Esperanto club to see if they know of any research or are willing to translate any to English.
There should certainly be some kind of research, as in general a lot of the Esperanto research focuses on kids (ex. how fast do kids learn it, how much does it help them with subsequent foreign languages), but likely it's only small studies. There's a few YouTube videos of kids who are native speakers talking, where it is assumable that their parents aren't native speakers (I'm thinking of one compilation video in particular, of kids of various ages and from various countries). The only study on native speakers I can think of has been where this one guy interviewed them and noted down what kinds of mistakes they make and how they use the grammar; a summary is readable on Wikipedia. Lastly you can pretty easily find older native speakers on the internet and simply ask them about how they were raised, how their skills were, if they know of any research etc (that is, if you learn Esperanto you can ask them).
About only speaking your native language with your kids... It's pretty easy/fast to become so good at Esperanto that it FEELS like your native language, and it's certainly possible to reach native-level fluency in foreign languages in general, so honestly I don't think it's a problem. Either way I wouldn't quite worry about it, as there are plenty of parents who speak languages other than their native ones to their kids.
About the logical-ness as commented in one of these comments here, it's been mentioned quite a lot that Esperanto is better for teaching/learning logic than languages such as Latin and English are — this mostly translates to that people think it's a good language for science, math, and computer programming (as in, it's good for testing ex. programs which parse language information). Again I'm not sure how many studies have been on it or that are described in English, if you go to the "open library" website there are a few books that you can read for free that describe various studies and general information. Though the books on there are older, the info is still valid since the language has hardly changed since then (the structure is the same, there's just more words now). I think most of the research is actually published and relatively difficult to get a hold of, as it would need to be purchased or ordered.
Sorry I can't help more. It seems like most studies of Esperanto tend to come from Esperanto users themselves, and they don't have enough money to do many studies. There are even fewer studies on native speakers, probably in part because most people who don't know about Esperanto assume that there are no natives. Although (and I'm not actually joking here), if you want to become famous in Esperanto-land all you have to do is make some studies or reports of your own on native-speaking children. You could start doing this before you have your child and see what your research suggests!