Not sending save the dates to people who might expect one

So, I have a dilemma and wanted to hear your thoughts. We are about to send save the dates and are considering cutting down some of our guest list beforehand. We are about 5,000 dollars over budget on our food and beverage. We would never dis-invite guests that we verbally told would be invited. However, what are your thoughts on not inviting people that you initially asked for their address? I directed people to a link so I could gather mailing addresses but never really mentioned what it was for. I’m also assuming that everyone already knows what I would need their address for so I’m worried they will feel offended. Should I cut my budget elsewhere and invite the guests that sent me their mailing address?