We are trying to keep our guest list at around 175 people for cost reasons (it's the most we can afford to feed), but we certainly could have more guests -- as long as they didn't eat. Is it okay to invite people just for the dancing portion of the reception, but not the wedding ceremony or the dinner?
Some ideas make sense on paper, but in practice...let's put it this way, would you want to be one of the people who showed up after dinner? While it is permissible to only invite some guests to the reception (keeping the ceremony intimate), inviting only some guests to the second half of the reception is a major faux pas. Just invite those wedding guests you can afford to include in your entire celebration.
If you really would like to have more people, look into ways you can make it work within your budget. For example, you might not be able to afford a seated multicourse dinner for 225, but you may be able to make a light buffet, passed hors d'oeuvres, or a luncheon work. Talk to your reception site manager or caterer about options to see if anything can work with your expanded list and set budget. A little creativity can go a long way in having the wedding you want.
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