It sounds like you guys are asking great questions and doing very good research. We have family in Idaho and are there often.
Yes, check out WPI. We got to know lots and lots of schools and that really was the family favorite. Like I said it often offers merit aid, and a high-stat, well-qualified female applicant from Idaho is probably a great candidate for that aid. I obviously don’t know for sure, and it would be best to run the Net Price Calculator (NPC) for the school (and others you are considering), but I’d really guess she would not be asked to pay the “sticker price.” Visit if you can; it is far away. But you can make it a nice family trip, if possible financially and time wise. It’s near Boston to the east and the Berkshires to the west. Cape Cod also. A very nice area in late spring and summer.
Are you familiar with the Western Undergraduate Exchange? It offers reduced OOS tuition at something like 160 schools in 16 western states, including Idaho. Some certainly are good for mechanical engineering. I’m not expert on WUE, at all, but I believe that some schools, at least, reserve a certain number of spots in incoming classes. Once they are filled, that’s it, so applying early is really important.
I’m less familiar with Mines and Tulsa, but have researched and been to both, and they definitely also seem like great choices, given your parameters. The others as well, and you are correct to look at that group as reaches. I always think it’s important to understand that they are reaches not because your daughter is not very well qualified, it looks like she is, but because so many more very well qualified students apply than can be accommodated.