I've been having the same issue. Just for fun, I switched my fluid block to extend from Vanilla's BlockLiquid rather than BlockFluidClassic, and to my great surprise, it fixed this rendering issue.
There's a method in net.minecraft.Block, getRenderType that returns an integer value based on the rendering type that Minecraft should use to render the block. In BlockFluidClassic's superclass, BlockFluidBase, this method is overridden to return a value from the Fluid Registry, renderIDFluid. The value defaults to -1. I don't see anywhere this value can otherwise be set (since there's no registerFluid method that sets this value for the fluid anywhere I can find), but this makes Minecraft not render the interior faces of the fluid block.
By overriding this method in my fluid block implementation to return 4 (which is what Vanilla's BlockLiquid class does), I managed to fix the rendering issue, and now the interior faces of the fluid block display properly and with the correct texture.
If a veteran knows the "correct" way of setting the Fluid's renderIDFluid value for mod fluids, please step in!
IMPORTANT: This will cause a crash if your fluid's Material is anything other than Material.water.
EDIT: I just looked again and noticed that renderIdFluid is a static value, so obviously cannot itself be changed per fluid block. Looks like overriding getRenderType is the way to go, but doesn't seem to work for custom fluid materials.