And I bet you didn't think twice about walking away for using them again?
This scenario (quote too low, end up over delivering) is a common occurance. The amount of jobs I've done that have been twice as long as they should have, because something is more difficult than expected, or the suppliers have increased their prices without informing me, is quite amazing, especially in the early days.
However I would never go back to the client and say "sorry I cocked up, the bill is actually 20% more". If I did, I would lose that client, or at least give them a very good excuse to look elsewhere. Just because the invoicing software cocked up, don't pass this onto your client. In your situation, I would pay the VAT personally, suck it up, and put it down to experience.
If you're clever, you'd try to turn a negative into a positive. Explain to your client what has happened, say you'll take the hit as you want to build a good relationship with them, and in future you'll be clearer on the total costs. They might just respect you a little more for that. By demanding 20% you're burning bridges