It is readily apparent that the two large banks face very different risks, since the second bank’s funding will at best become more expensive and at worst decline or disappear. A sound regulatory regime will align the riskiness of the two banks more closely, by requiring the second bank to change its funding profile, increase its capital, or some combination of the two.
In fact, the LCR and NSFR would require more stability in funding at my second hypothetical bank, while one of the factors used to calculate the capital surcharge applicable to the largest banks is its relative reliance on runnable funding.