(Some of these may be repeats… I didn’t go back through all the posts.)
These are some schools often on the application lists of full pay families seeking merit. These do not all offer engineering, but they are all good choices for premed. This list is by no means exhaustive but an example of the many options your D has.
IMPORTANT: Some schools require earlier due dates to be considered for merit so that needs to be looked at asap.
If your student has preferences in size, urban/suburban/rural, Greek life, sports, etc., we can help with a more targeted list. I hope I haven’t missed that information in my quick skim of your posts. 
LACs - Rhodes, Wooster, Macalaster, Denison, Dickinson, Gettysburg, Furman, Trinity, Whitman, Kalamazoo, Juniata, Bucknell, Lehigh, Lafayette
Mid - Case Western, SMU, Miami-OH, Rochester, UAlabama-Birmingham
Large - Pitt, UAlabama-Tuscaloosa, UMN, SUNYs
Those are all very realistic for admissions and getting merit (or have a lower COA for OOS). For more competitive merit options, if your D is interested in the extra work involved and early due dates, look at:
Vandy, WashU, Davidson, Wash & Lee, UMD, Miami (FL)
Many families must consider their student’s post grad goals and choose to save on undergrad. There are many posters here who can help with suggestions. Best of luck!