Can I bake oven-bake clay with embedded lobster clasps?
&quot;Can you&quot; is a different question from &quot;is that the best way&quot;.  I think the answer is you can, but it probably isn't the best way.
Embedding the clasp, then baking
The metal in the clasp shouldn't interfere with the clay's hardening or properties.  The clasps don't typically have a finish that would be damaged by the heat, and even cheap pot metal shouldn't melt at the clay's baking temperature.
One potential issue that comes to mind is that metal expands and contracts with temperature more than the clay once it's hardened.  Cold isn't likely to be a problem.  While the charms are baking, they will be exposed to far higher temperature than they will see later.  So expansion won't be an issue.  But when the charms cool after baking, the clasp will shrink slightly and might be looser than you expect (captured but not &quot;bonded&quot; to the clay).
I see a bigger potential issue being ease of breakage.  You mention that you're thinking about selling the charms.  If you're making them just for yourself, you can be very careful how you handle it.  If it breaks, you can make another.  If you sell them, customers will be disappointed if they easily break, and a reputation for quality will help your business grow.
There will be a lot of leverage at the attachment point in the clay.  Lobster clasps are normally attached with a ring, which allows the clasp to move in any direction without stressing what it's attached to.  If the clasp is embedded in the charm, the clay will be much thinner around the clasp.  If the charm is thin, it won't take much pressure on the clasp to break the clasp out of the charm unless you build up the clay thicker around the attachment point.
Attaching the clasp after baking
As mentioned above, these clasps are designed to be attached with a ring, which leaves the clasp free to move without putting stress on what it's attached to.  You can drill a hole in the charm for a ring after it's baked, or make a hole in the clay with something like a toothpick before you bake it.
If your objective is to have the clasp rigidly held in the charm, gluing it to the surface of the charm isn't a good solution.  I wouldn't trust hot glue, there are better adhesives.  However, it won't look good (and you're thinking about selling them).  If you want a rigid attachment, you might as well bake it in initially, but with building up the attachment area so there's a reasonable thickness of clay around the clasp.
You might find complications doing it that way, like getting the clasp to lay parallel because of differences in thickness.  You could play around with finding something the right thickness to shim under the clasp or charm for alignment while it's baking, or you could make a thick-walled pocket in the clay to glue in the clasp after baking.
If you go that route, hot glue probably wouldn't be the best way to hold it in.  The pocket will be tiny, and it will be hard to get hot glue into it, at least without messy cleanup.  Epoxy would commonly be used for this, but I'd be tempted to use something like clear silicone adhesive/caulk.  That will remain rubbery, which will provide a little additional protection against breakage.
Gluing procedure
If you glue in the clasp, it will be hard to get the glue to flow into the clasp's ring loop because there probably won't be much clearance between the clasp and the sides of the pocket.  Use a procedure like this:

Fill the pocket half to two-thirds full with glue.

Depending on the size of the clasp's loop and the viscosity of the glue, use one of these two methods:

Fill the clasp's loop with glue, then slowly push the clasp into the pocket.
Or, push the clasp's loop halfway in, fill the loop starting with the leading side so air isn't trapped, then continue to fill the loop as you push it into the pocket.


This will force the glue in the pocket into the nooks and crannies and allow air to escape.  It will also displace some glue from the pocket that you can push back in to backfill around the clasp at the top of the pocket.  You may need to add more glue to the top of the pocket once the clasp is all the way in.

Wipe away any excess and clean up the charm before the glue cures.