All are viable solutions. I would recommend combining 2 & 3. Ball and burlap and then heel in a temporary location. Then move to the new location when it is ready. This minimizes transplant shock both times.

A couple precautions.

When heeling in a plant, make sure it is mulched real well so that no burlap is exposed. Burlap wicks moisture out of the ground.

Areas near foundations of additions tend to be alkaline from the mortar, cement, etc. You may need to remove any such contamination or use a raised bed. If your soil is naturally acidic, then you may be able to just apply powdered sulfur for a couple years.

If you use natural, untreated burlap, you can leave it in the ground after opening up the top and making sure it is covered with soil and mulch.

If you use treated burlap, you will need to remove it when you plant them to the final spot.

Here is a link that might be useful: How to care for azaleas

Are you about a zone 8 there? If you you think you can really be ready in a month or so, I think you are safe going with the burlap. Try to take as large a rootball as you can manage, move them someplace protected out of wind and sun if you are having sun. (rough rule of thumb is aim for a rootball equal in diameter to 2/3s the plants height, even though you can expect roots to extend out as far as the widest branches)

Piling up some soil or mulch temporarily around the burlapped roots will help to retain moisture and protect them from cold - that may be less traumatic than installing them in the ground briefly, and you could run the risk of temporary planting holes filling up with winter rain and staying too wet too.

I hope your renovation goes smoothly and on schedule. We burlapped several rhododendrons for a neighbor one year for a remodel project that somehow became an all summer into fall job. She did lose about 3 of the 8 but I suspect they had very little attention all those weeks.

My son in law bought his grandmothers house and had some very large Rhodies up against the house the ground was very hard so after digging to disgust he put a chain around them and pulled them out with his truck.Replanted them in a better spot and they are awesome after reestablishing.

In 2006 we had our old doublewide replaced. I had a landscaper move 6 mature azaleas, 1 10' dogwood, 2 hydrangea, 1 Bloodgood japanese maple, and 2 pieris japonica. It was done at the start of July (I had no choice). We created a holding bed in an empty planting area, covered the soil with straw mulch, and watered the living "tweedle" out of them. We were unable to re-relocate them til 2007 but did not lose anyone. I enjoy caring for all my "tough guys". Just got some Pine Straw mulch for them. They deserve to be spoiled.