I received a Schedule K-1 that has Box 16 checked (K-3 is attached), no Sch K-3 attached and the Instit that sent the document says the K3 will not be available until July 1. Should I unchecked the K-3 box right now, file by mail with a 8082?

Filing for an extension is the safest option.  However, there is an exception stating in 2022 regarding K-3's.

New exception to completing Schedule K-3.

“Under a new exception, in certain cases, partnerships with no or limited foreign activity are not required to furnish Schedule K-3 to you for tax years beginning in 2022. See Domestic Filing Exception in the Partnership Instructions for Schedules K-2 and K-3 (Form 1065)."  New exception to completing Schedule K-3

Also see:  Domestic filing exception (exception to filing Schedules K-2 and K-3)

If you choose to uncheck the K-3 box, I suggest filing a Form 8275.

Form 8275 is used by taxpayers and tax return preparers to disclose items or positions, except those taken contrary to a regulation, that are not otherwise adequately disclosed on a tax return to avoid certain penalties.

Form 8275 is not supported by TurboTax, and it cannot be Efiled.  However, you can mail your return and use the link below to prepare the 8275.
Please see About Form 8275, Disclosure Statement for instructions and the FORM.

 Your Federal tax return goes to a different location depending on the state that you live in.  See this list to determine the location for you. Where to File Paper Tax Returns (irs.gov)

  If this does not completely answer your question, please contact us again and provide some additional details.