ArkInventory does something clever, they send things to the reagent bank using a PreClick handler and so they don't have to do all that annoying messing around with the Blizzard bank frame and it's selectedTab so that the OnClick handler works.

It's basically this, which I am stunned works:

function LiteBagItemButton:PreClick(inputButton)
    if InCombatLockdown() or not self:HasItem() then
        return
    end

    if LiteBagReagentBankPanel:IsShown() then
        UseContainerItem(self:GetBagID(), self:GetID(), nil, true)
    end
end

Edit: I think this has a downside that if the item does not go into the reagent bank, the OnClick handler will put it into the regular bank which isn't currently visible.

Too risky with taint, not gona do it