I'd suggest that, rather than try to track the individual customer, that the fast-food places be obligated to prove (by whatever means they like) that X% of their sales transactions have the packaging disposed of properly. Obviously, 100% would be an impossibly high target, but even a relatively low figure (say, 30%) would, I feel, force businesses to be proactive about rubbish collection so they could point at all the stuff they've collected as part of their target! Plus it would encourage the businesses to offer reusable serving materials, because those would be sales-without-packaging, so a "free" point on the scale.
And then you could slowly ramp the target up, and possibly use local council licensing to maintain a range of targets depending on location, so that black spots for litter would have higher standards until they're cleaned up.
A side benefit is that it would likely result in the fast food chains employing people to pick up trash, so they could meet their rubbish-collection targets as well as their profit targets.
Enforcement could be via local taxes: exceeding the target demonstrates social responsibility, so naturally that business is more valuable to the community and deserves to pay lower taxes, while missed targets label the company as a drain on society, deserving of higher taxes.