Have you checked the manual?
From the pictures I could find online, the ASIC looks like it only has a couple of precision resistors in it.
The one with golden traces on top, it's similar to those you find like on HP-3456A:
http://postimg.org/image/kc0y9l483/So my guess is that if the manual has the resistance values of ASIC listed (HP-3456A manual has them listed) and since HP-3478 is only 5 1/2 digit, you could use a couple of precision resistors (like the ones from Vishay/ Alpha Electronics) and replace the burnt ones inside the ASIC with discrete ones soldered on top.
BTW, I have a big box filled with various more or less precision resistors. I you tell me what you need, I can send them to you, in exchange for rounded up shipping