I would think that your bank would be willing to give you a card, a secured card at the minimum, as you have $12K on deposit with them!
To establish a FICO score, you need to have these things:
- An account that has been open for six months or more
- An account that has updated (reported to the bureaus) within six months, even with a $0 balance
- You can't be dead. Or reported dead.
Once you get a card from your bank, give it six months, and you're good. If you need something sooner, you can ask family members if they would be willing to add you as an AU (authorized user) to one of their cards that reports AU's to the CRA's. You want a card that's been open for more than six months, obviously, and one without lates, and that reports a very low or $0 balance. If they use the card a lot, let the balance post, and then pay, that card might still have high util.
But at any rate, it would establish a credit history for you.
While you're waiting on your bank to decide, I would look into local credit unions. If your bank turns you down, politely tell them that you will ever-so-sadly transfer your accounts to Credit Union X, Y, or Z, and see if they don't reconsider. 
I hope that you're with a local or regional bank, and not one of the Big Boys that really don't care much about consumers any more.
You don't have a score of 0 (that doesn't exist in FICO scoring), you just don't have a score.
Most folks who are starting out with one card will have a score of between 680 - 720 after six months of perfect payment. If you pick up a second card at the six month point, you can probably expect to be in the 720- 740 range after 12 months, as long as you never screw up and if you keep your reported util down.
It is much, much, much, much better to be starting from scratch than to be repairing past errors. You won't have any negatives dragging you down for years.
It doesn't matter how large or small the CL is; you just have to stay under 10% of the credit limit <--as in 9% or less. So with a $500 CL card, don't let it report over $45. With a $1000 CL card, don't let it report over $90. And so forth. It's fine to let it report $0 as well. You're not trying to tweak at this point; you're just establishing a solid track record of credit card use AND payment.
There are plenty of other posts here on the credit card board that advise how to time your payments to control what reports. If you do wind up with a card from US Bank, be aware that they report the balance on your account as of the last business day of each month, reporting that night. This is different from most cards that update on the statement date, so just keep this in mind when you read other posts about when to pay, etc.
IMO, the main thing to do with any CC is to always have enough cash in the bank, or money coming in on the next paycheck that isn't already spoken for, that you can pay off your entire balance at a moment's notice. Do NOT use a CC to actually buy anything on credit. Just think of it as a slow debit card, needing money in the bank to back it up, which will be paid off by the end of the month.
Good luck on your app!
To me, having good credit is similar to insurance. You should rarely have to use it, but if you want to buy a house or a car, and you need to finance, it's great to get the lowest rates going. That's just good common sense.
But as for the day-to-day use of credit cards, many of us have found that they work best as tools to let us pay our bills once a month, or earn rewards, or get special deals from stores, or maybe once in a blue moon finance something at 0%. We just don't let them sucker us into buying Stuff that we can't pay for right away, and really don't need in most cases.