Hi Eric, I'll try to answer your further question even if I'm not an oracle like Pete
First of all thanks Pete to underline that you need to give google just one response since you can't give them both 301 and 404, I was assuming that and I didn't focus on that part of Eric's answer.
Second. Eric, If your purpose is to give google the ability of recrawl the old content to let them see it has disappeared you want to give them a 404 or a 410 which are respectively not found and permanently not found. Before it was a difference but now they've almost the same value under google's eyes (further reading). In that way google can access your page and see that those contents are now gone.
In the case of 403 the access is denied to anyone both google and humans, so in that case google won't be able to access and recrawl it. If your theory is based (and I think you're in the good way) upon the thing that google needs to recrawl your content and see it ahs really gone, 403 is not the response you should give it.