@Ethan Smith. You can always sell as yourself and buy as yourself and then contribute the property into a new LLC. That's no problem.
But, If the LLC is a disregarded entity (meaning that its only members are you and your wife, you file a joint personal tax return, and the LLC is taxed as a sole proprietor so it doesn't file it's own tax return) then you can sell as you and your wife and purchase as that LLC anyway.
The key is that the tax payer has to be the same for the old property and the new property in a 1031. The tax payer as the iRS knows it is really the tax return that reports the activity of the property - not whose name is on deed. So if you and your wife report the activity of the property on your personal return. And the LLC will not file it's own tax return. Then the activity of the property will still be on you and your wife's tax return. The tax payer will not have changed. That is fine for a 1031.
Whether you should or not??? Well it's not going to be $3K in tax. It's going to be probably $30K in federal, Another $15K in state, and whatever depreciation recapture there is which will be at 25%. That's not a small amount.
So, given the amount of tax, the low cost of a 1031 (here's a BP blog we wrote on it - https://www.biggerpockets.com/blog/how-much-does-a-1031-exchange-cost) and the ability to place the new property into a disregarded LLC I'd probably do it if it were me.