I had another thought @theallan as I re-read the thread and your statement about "nothing in it for them" - it's a long thought so stick with me.
We are all assuming their is value in the stamp of approval that BIGCO LOGO brings. That value could literally be $1 or $100,000,000, but we don't know. We can't figure out how to unlock that value appropriately, partially because we don't know what it is worth. If I knew that every BIGCO LOGO brought me two more enterprise clients worth $100k in ARR, I'd fight a lot harder to put it on my site, my guess is you would too.
All that said, is there a third option where in the contract, you define that value and let them decide what it's worth to them? I'm thinking something along the lines of in your contract you create an opt-opt for the logo inclusion, "In the event that you do not want your logo included, we offer to opt you out for a fee of 10% of the annual contract" Obviously pick your own numbers, maybe it is a flat fee instead of %
You get (some of) the dollars you hoped for with the marketing boost the logo would provide, and the BIGCO gets an easy way in your contract to decide what it is worth to them (and no one has to fight existing rules of endorsement). Additionally, you could of course waive the fee at any time for any company you wanted as a point to close a deal.
The inverse of this would be to charge more and then offer a co-marketing discount to the smaller companies who already are happy to be mentioned. You'd probably need to increase your prices so you could even drop this down to the normal pricing for anyone who opted into the program, but as above the outcome is still hopefully the same.
Of course any changes could change your close rate of business, and you may really just want the logos/endorsement on your site. But just trying to offer another alternative.