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CSR and Freedom Unlimited (if the extra 1/2% is worth the new account = 500 more UR's than CSR per 1k spend) would be my pics. Congrats to you both on having great credit. Tell partner "great scores!"
I really cannot find a flaw in that report other than questionable utilization that can be quickly fixed with a CLI. Overall If you want a card you can pick and chose one or multiple specially if you make the higher tiered card application go through first. I cannot imagine any bank declining that kind of profile and most cards should be instant response too.
Chase should have a Request Credit Limit Increase button on their site.
Walmart/Synchrony definitely has one on theirs and you can chat if it says no.
AMEX I heard has one too but calling in might be the the only option if there isn't.
The button is great for those of us that don't want to deal with people.
Quickest thing would be some Credit Limit increases. This would decrease his utilization, and thereby provide an instant score increase, although the scores are very high already. Amazingly high, considering the utilization. Paying down balances would help too.
Emphasize that he should get the CLI with soft pulls, if possible. This is also a good time to ask for an interest rate reduction.
Other than that he will likely get whatever card he wants. The top 3 travel cards right now are the Amex Platinum, Chase CSR, and Citi Prestige. Right now the edge probably goes to the Chase CSR.
Good everyday rewards card: right now Discover IT would likely give him a high CL, and your first year of cash back they double on your 13th statement. They have 1% on everything, and 5% cash back each quarter on revolving categories (this quarter, Amazon - Dept Stores - Sams) for the first $1500 of spending on those items in that quarter. So you get 2% on everything, and 10% (up to $150 back) per quarter. After that, depending on spending, any 2% cash back card would probably be as good (Citi Double Cash).