This is kind of a "code mess" because this kiln manufacturer is good at making kilns, not so good at complying with NEC/UL rules, especially in 120V/16-20A current draw. As such, they unable to achieve UL/CSA listings in the US.
The hangups are twofold: first the device needs a 30A plug, and these are highly uncommon in US homes. And second, your dryer receptacle needs to provide both neutral and ground - and if it's a 3-prong Halloween outlet, it does not have that (possibly because the ground wire isn't there). Let's solve each, one at a time.
Your device draws 19A actual, and you need to provision power for 125% of that since it could run continuously. Therefore you are talking 23.75 amps, too much for a 20A circuit obviously, and you need to upsize to a 30A plug, socket, wiring and circuit breaker. This makes it a good fit for that dryer breaker and wiring.
That means you need to use a NEMA 5-30, or a NEMA L5-30, or the odd TT30 (intended for small travel trailers).
However, since you want to co-use this socket for your dryer, I suggest simply fitting a NEMA 14-30 plug. This is a 30A 120/240 hot-hot-neutral-ground plug, which is overkill for your needs, but will fit your dryer outlet. Quite soon.
This may get a little convoluted, depending on the age of your wiring.
If your dryer outlet is already a 4-prong NEMA 14-30, then you're done. Congrats!
Get a NEMA 14-30 dryer cord for your dryer. Install it according to the dryer's instructions. This will include a step to remove the bonding between neutral and ground. Very important, or it will defeat the safety protection we're about to add!
Now, pop off your obsolete NEMA 10-30 dryer receptacle and see the wiring behind. If a ground wire is present (but not hooked up), then install a NEMA 14-30 receptacle. You're done. Congrats!
If you don't have a ground wire, but can confirm the conductor wires run inside a metal conduit from the dryer all the way back to the panel, then look in the back of the junction box for a hole a bit smaller than the rest. It will take a #10-32 ground screw, and they sell green ones 10 for $1. Attach a 10 AWG ground pigtail to that, then install a 14-30 receptacle using that pigtail and the 3 conductors. You are done!
If you have the physical access to physically retrofit a 10 AWG bare ground wire between the dryer receptacle and the service panel, then definitely do that. Use that ground wire on a NEMA 14-30 receptacle and you are done.
If none of those are possible... Replace the dryer's circuit breaker with a 30A 2-pole GFCI circuit breaker. Make sure this is possible before continuing. Once that is done, install the NEMA 14-30 receptacle with nothing connected to the ground screw. Then, stick two labels on the receptacle (these come with the GFCI): "GFCI Protected" and "No Equipment Ground".
At this point you're all set for using that receptacle for both dryer and kiln.
Frequent use: Second receptacle
These large receptacles are not made for frequent use. This can break them. If you need to do this often, then get a "junction box expander" for the physical space, and splice on a circuit extension to go to a second receptacle. That way the dryer can stay plugged in all the time.
Since it is a separate receptacle, it can be any 30A rated type - NEMA 14-30, NEMA L14-30 (locking) or TT30. It does not need to be located right next to the dryer.
Yes, it's legal to have two receptacles on a 30A circuit. In recent years we researched the heck out of this, because most people here didn't believe it.
Big extension cords - avoid
There are three huge problems with big extension cords. First, if you get a long cord, you will be tempted to leave most of it coiled up. The extension cord will melt and catch fire if you do this.
Second, long extension cords are vulnerable to voltage drop, especially when you chintz out on them. For instance you are proposing 20A extension cords and your need to provision 24A, so you're running at hard thermal limits (worsening the "coiling" problem) and will experience voltage drop as high as 5%. That will reduce performance of the kiln.
Third, long extension cords are frickin expensive. The ones you linked are $2 to $2.50 a foot. Building hard-wiring is cheaper and higher quality by far.
"Hard-wiring" can include a pendant connection, which is stopping at a junction box then splicing to 10 AWG cordage with proper strain relief. The cordage can be any length, then end in the appropriate inline socket. The cordage must be 10 AWG, but if the socket is 3-conductor, the cordage can be too.