Define not a strong engineering program. Colby or Davidson or Haverford. Dartmouth - you go a 5th year - three at one school and two at another.
If you want to study engineering, you study engineering. My kid chose Bama over Purdue - at one he got his own room…the other, at the time, was overcrowding and housing kids off campus and he didn’t want that).
Find your discipline. If it requires ABET to get a job (like Mechanical) you make sure your school is accredited - and you’d be amazed how level the playing field is. Yes, a few schools will always stand out - but not necessarily those on this list (maybe CMU).
If the schools above don’t offer engineering - eliminate them. You have to get down to one so it really is that simple!!
As for extra tuition - it may not apply to any of these schools but it does apply to some. 17 hours in a week - so that could be 5 classes. It’s school and curriculum dependent. So I’d check just to be sure so you are aware up front.
As for job opportunities - you never know. You could be in Cleveland and find a role in Cleveland because you were local. But what I’m trying to say is - kids are finding jobs from all over - so there could be advantage - but not necessarily. Mine had 20 interviews Fall Senior year and 5 offers. School was in suburban Alabama - and the offers were Raleigh, Massachusetts, and then three where he’d have moved all over - one in the rural midwest, one with a chemical company in the south and midwest, and the one of which he took - so far Utah and Arizona and next will be California. Not sure of the last but maybe Ohio where the company is. He chose to be out west…he could have chosen east coast locations. He’s in a leadership rotational - so the first two years are 6 month stints. He interned two summers at an automotive and they came late with an offer (as they told him they would) but he preferred a rotational program vs. a standard job and applied only to those types of jobs.
Gotcha on costs - glad we are in the same place 
You know - college costs A LOT - and so that is personal to every family.
Cooper Union is hundreds of thousands less than the others and still will be - after high priced R&B. It’s ABET accredited in Chem, Civil, Electrical and Mechanical. So you are willing to pay - but do you really want to pay? Get with them to check their career outcomes.
Then you say - ok but it’s very small, quirky and the like - so is that the right fit?
To me, your choices are - and I don’t know the discipline:
CMU
Cooper Union
CWRU
Lafayette
You can’t go wrong with any - and there’s some like my son - his “merit” was based on stats and I spent maybe $70K all in over four years - if that.
One could make a choice for any of these four.
CMU might be the biggest name and Lafayette the smallest name - but that’s not necessarily going to be an outcome difference. You might find “outliers” salary wise at CMU - because it is a true top top name - but those are outliers, etc. and you just list general engineering, so maybe the student isn’t yet sure. At CMU, many end up in finance jobs hence the outliers.
Have you been to all? Part of it would be - I’m here - where do I feel comfortable?
The Cooper Union (small, NYC) and Lafayette (a suburban LAC) will be the outliers here - but your student might feel great, at home at one of them.
The flipside CWRU is close to downtown and little Italy - and my kid hated the campus - but on the college confidential, it seems to be near unanimous love for everything about the school including the campus. CMU seems more a fit school based on what others write - so it’s important that your student fit there - it goes beyond academics. You are there four years, day after day after day - so you want to be where you want to be.
But I don’t see why you would consider a non-engineering school of this list.
You have to get the list down - so that to me is an easy cut. Nothing wrong with Davidson - but I’m not going there if I want to be an engineer, etc.
Dartmouth - I don’t consider because their ABET engineering is planned over 5 years - although they say it can be done in four - so maybe look into that. It’s rural…but again, their plan is 5 years - so it’s not a 3/2 like Colby/Davidson/Haverford - but it’s a longer path.
Good luck to your student - but visit all and see - where do I feel at home!!! To me, that’s step one!!!