Your plan is reasonable. I have made many components for baskets by steam bending relatively thin slats, then glue laminating them together to get thicker, curved and very stiff pieces. I would offer some suggestions for how to get to success (and will answer your question about drying along the way).
First, selection of your wood is important. You want strips with no grain runout at all if possible, so you want to cut the strips from quarter or rift sawn lumber, or from the straight grained edge of a wider flat sawn piece. I would go with three 1/8" strips, rather than two 3/16" for your work. Air dried is better than kiln dried, but as you probably can't get any, with strips as thin a 1/8", kiln-dried will be fine. 1/8" strips will be easier to work with, and having two glued surfaces in the laminate will actually make a structurally sounder and stiffer final handle. If you want a 1 1/2" handle, as you suggest, make 2" strips. But structurally, a 3/4" or 1" handle would be just as good.
Make yourself a form over which to bend the strips. In your case, that will be a 2" wide form that fits within the inside curve of the handle. You'll want to make sure you build the form so that you can clamp the strips to the form all along the length of the handle (and yes, you'll need quite a few clamps - for cuves like this, a clamp every 3" is not excessive).
With 1/8" strips, the steaming can be done in a number of ways. You can steam the strips in a steam chest (or a plastic bag) with steam from an electric steam generator, or a self made generator (saucepan with lid, hose, and some silicon tape), and when the wood is pliable, clamp it to the form. Or you can just clamp a strip to the center of the form, direct steam at the thin strip and slowly bend it around the form, steaming and clamping as you go until you reach one end, then do the other side the same way. I've used this technique many times for one-off laminates with tight radii. You do want good gloves and to take real care around the live steam, though. Live steam will cook your fingers very quickly.
Once you've got a strip steamed, bent around the form, and clamped, leave it for at minimum several hours. When you remove it, it'll spring back some, but that's not a problem. Tie the ends together with a long strip of tape or something, and let it dry.
You can bend all three strips on the same form in separate operations, or bend and clamp them all at once. Either way works.
I would let the strips dry for a day before gluing. More won't hurt, but won't really help either.
For the glue up, you want to coat both sides of the glue line with glue (I use waterproof PVA glue like Titebond 3 for these, but anything that gives a reasonably hard glue joint and has good open time will work; Epoxy has the advantage of a long open time, but can be very slippery and messy to work with, whereas a thin, but complete coat of PVA glue tacks the strips together making clamping easier. But, if you prefer it, epoxy is structurally a good choice). Make yourself a fourth strip and cover one side of it with plastic packing tape, to use as a caul and clamp pad.
For assembly, put the glued strips on the form with the clamp pad on top, and start clamping at the center (top of handle), working both ways to the ends. As I say, you'll want clamps every 2 or 3" along the form, with short block cauls where there are reasonably flat stretches on the handle profile. Your biggest challenge is getting the glue joint tight (thin glue line) as you go around the curves. If you've got a strap clamp, one solution to this challenge is to use it to clamp the strips around the entire form (so you have a layer cake of form, 3 strips of glued wood, the taped caul, then the strap clamp). Then use regular clamps right over the top of strap clamp to thoroughly seat the joint.
As an alternative to the above, you could also cut yourself even thinner strips, and skip the steaming - just glue laminate them. You might get by with that with 1/8" strips too, but the radii you're after are probably at the margins of what you can get in a 1/8" strip with no prebend.