You will be hard pressed to find specifications for the ceramic plates
used for peltier devices and the most I have ever found is that they are
made with alumina ceramics. Note that alumina ceramics have different
properties than pure alumina.
In general, alumina ceramics are weakly attacked by strong acids and
bases and insoluable in water.
A ceramic that is 74%, which is about the cheapest stuff anyone makes,
or more alumina is concidered gas tight and will not absorb water.
I would doubt ions could migrate through alumina ceramic.
Some random thoughts on the issue.
Commonly available peltier devices are 40 X 40 mm and you will need a
heat sink on one side of it.
Unless you have a short length of glass tubing with a good, square cut
end you can somehow glue to the device, the simplest thing to do is to
themal glue a small beaker or flask to one side and the heat sink to the
other. Keep the glue layer as thin as possible by applying pressure
while the glue cures and do one side at a time.
You will want some thermal insulation around your container.
Peltier devices are current controlled, not voltage controlled, and are
best controlled with something like a cheap motor control module from
Amazon.
Switching from heat to cool is trivially done with a DPDT toggle switch.
Air cooling is a PITA so you should concider a 40 X 40 aluminum water
cooling block (about $5.00 on Amazon).
Amazon also has cheap pumps for around $15 you just put in a bucket of
water as well as pump systems with attached tanks starting at about $40.
With a thermistor thermal glued to the container and a few parts, you
can make the stuff to drive the motor controller such that the
temperature is kept to fractions of a degree C.
Such circuits can be found on the internet and the parts are, again,
available from Amazon.