Do not complicate a simple matter.
I0ser20:
To reiterate and paraphrase Army Judge's opener, the words "grace" and "visa" do not belong in the same sentence and as such there is no grace period for renewal of an expired visa, not these days anyways. It used to be that prior to 1998, an alien could simply walk into an I.N.S. office and renew his visa after it had been expired for months without batting an eyelid, alas no more; not these days.
An expired visa will subject your boyfriend/fiancé to arrest, detention, and removal proceedings, and he will also be restrained by a 5 year stay away (from the U.S.) period. So it would stand to reason for your fiancé to address an extension with I.C.E. or U.S.C.I.S. before June 20, 2011.
But in my opinion, by going for a K-1, K-3, or whatever other temporary status visa there is available, you will only manage to complicate a simple matter which is capable of being dealt with efficiently and put to bed at the first time of asking instead of dragging on for eons with uncertain results looming tall. One such complication would be the obvious chasing of a meaningless temporary K-1/K-3 visa that is bound to sap your energy, wits, and finances to the max. Another potential monkey wrench in the works would be the implication of your marriage ed in a few months when your fiancé will perhaps be in the throes of fighting for his immigration life.
In other words, your marriage in the future and potentially during a very tenuous period may, and I emphasize the word may, come across as having been entered to for convenience and in order to "circumvent" the immigration laws of the United States.
I therefor really thing that THE best next move for you two would be to tie the knot now and then by submitting an application to the I.N.S. for Adjustment of Status to an L.P.R. (Lawful Permanent Resident) by way of marriage to a Citizen. That way, he will enjoy some of the instantaneous and automatic privilege of the application such as automatic work authorization and an undisturbed Leave to Remain in the U.S. until the final outcome of your case.
Good Luck!
fredrikklaw