The armor is described as being plates on lacquered leather bound together over a silk shirt. Couldn't the plates be made with mithral instead?
A Lamellar Cuirass, no. That is specifically "made from lacquered leather plates bound together and fitted over a silk shirt."
Other Lamellar Armor can be made from metal. Specifically, Steel Lamellar armor is basically like Breastplate, but with a higher ACP. Also Iron Lamellar is basically Splint Mail. That could conceivably be made from Mithral too, though it's maybe not necessarily clear how it works since the AC benefit of the armor is based upon the material used in its creation.
Since Mithral is "lighter than steel but just as hard" the Mithral Lamellar (~Breastplate) Armor should work. The Iron (~Splint Mail) version is less clear, but it probably wouldn't be a huge deal to allow it, since they're both fashioned primarily out of metal.
Lamellar curiass doesn't appear to use any metal in the original version so I would say no, you can't make it out of mithral.
Why not just wear a mithril chain shirt?
Making a mithral lamellar cuirass is equally meaningless as making mithral leather armor or mithral hide armor. Fortunately, the devs gave us a rule for this:
Items not primarily of metal are not meaningfully affected by being partially made of mithral.
So, technically, I suppose you could make parts of a lamellar cuirass out of mithral, but it would not be "meaningfully affected" if you do - so it would have the same statistics as if you just use lacquered leather.
There's no question you can't get any benefit from a Mithral Lamellar Cuirass. But the question would still be up in the air for Lamellar made out of steel or iron. It depends on whether people think they're primarily made out of metal or not.
The reason I'm asking is to see if I can lower the Arcane Spell Failure of any armors an unproficient witch could wear.
Why not make your Lamellar from Darkleaf Cloth then ?
Unfortunately, darkleaf cloth only reduces it to 5%. If there was a way to lower it further...
How about mithral darkleaf cloth?
...
j/k
You can take arcane armour mastery if you really want that high armour low spell failure armour (and don't mind burning swift actions).
How about a different tactic: Bracers of Armor - all the armor you could ever want (more than lamellar cuirass (fully enchanted you won't get more than +7 AC but your bracers can give you +8) with no DEX penalty and no Arcane Spell Failure - and it works against ghosts, too.
Arguably, some day he might want +5 lamellar cuirass of X, Y, Z and a few other enchantments too, adding up to a +10 item maybe with a couple other add-on powers that raise the price but not the effective enhancement bonus. Fine. By the time he buys such a thing, he could invest in the feats or level dips he needs to ignore arcane failure anyway, for example, get the feat Arcane Armor Training and just get a mithral shirt (more AC, zero arcane failure if you spend your swift action for it).
Note that he might also need/want to find some way to get Light Armor Proficiency - it's required for the Arcane Armor Training feat but not necessary otherwise, though any spells he casts with an attack roll (touch attacks, rays, etc.) will all suffer the armor check penalty if the armor he wears has one. Mithral chain shirt does not.
For all it's worth silken ceremonial armor has only 1 AC less than the lamellar curiass but a higher maximum dex and no arcane spell failure chance. Why not just have your character wear that?