I think this depends on the number of people coming. If you are sending printed invitations for the dinner, it would indicate that it is a hosted dinner. Since that is not what you are planning, perhaps the invitation could be via word of mouth, Evite, or something less formal.
Perhaps your family could pass the information along to each other. You could also ask your maid of honor or best man to do the inviting - that way it won't be assumed that you and the groom are hosting.
Really, my best suggestion is to take your parents and attendants out on another day, and invite guests to something you can afford. You could have a cake only reception - even cake and punch with a cash bar (or no bar). You could do a create your own dessert bar, or even a potluck style dessert bar if your family would be willing to help.
This way, everyone can celebrate, you don't feel rude, and guests are not surprised. Good luck!
Check out this site for the dessert bar idea and how to:projectwedding.com/wedding-ideas/budget.
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