I'd try calling at different times of day and on different schedules. The hope is to try and reach either:
a) a cool rep
Or
b) reps on a different shift with a cool supervisor

Perfect situation is to get both. Politeness is essential.

I do think the supervisors have some leeway in being able to offer retention bonuses in spite of whatever the frontline rep's computer says. The key is getting the rep on your side so they'll go to bat for you with the supervisor. It might be worth mentioning to the rep that you've put X spend on the card and you'll put Y spend more. Also telling them you're thinking of switching to Z card (politely) with more relevant benefits could help. I know it takes some work, it's kinda like arguing a recon call, or going to court even.

That being said, from my experience Citi (especially), Barclays, and Amex tend to be more generous with retention offers. I know Chase is taking a whooping from the Amex BBP and Personal Gold right now. All one has to do is look at the increased bonuses on Ink cards (with a coincidental $50K spend MSR...hard to figure that out lol) and to a lesser extent the CSP. Citi Thankyou cards are also getting more competitive. And of course Amex MR transfers to British airways (they've got a 40-50% transfer bonus right now). Don't know if those other cards are relevant to your situation, but info like that could help you construct your argument to Chase.