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@Saeren is correct about your FICOs being on the low side to obtain the starting limits you are looking for. Financial institutions use your FICO scores as a snapshot of your current creditworthiness however it is not the only criteria used in making lending decisions.
Ultimately, what a person needs to improve their FICO scores and build credit are three open credit cards (secured or unsecured) in good standing and one open installment loan in good standing such as a car, home, student, personal, share secured, or credit building loan. This combination is what the myFICO score theorists here have determined is what you need for optimal credit building and FICO score. You can have more CCs and more installment loans, however, this will not increase your FICO scores.
Next, (this is only important if you are attempting to purchase a large ticket item like a home or car) is paying in full all of the credit card balances each month, before the posting date, except one. This is called the All Zero Except One (AZEO) method. The one credit card you allow to post a balance needs to be less than 8.9% of the credit limit of the card. So using one card each month to buy lunch, letting it report and then paying in full will maximize FICO scoring. Keeping your utilization of your cards below 28.9% both individually and collectively will prevent you from incurring a FICO scoring penalty.
An installment loan will have its greatest impact on your FICO score when the amount owed is at 8.9% or less of the original amount owed which is usually in the final months before the loan is paid in full. If you don't have an installment loan you can check into Self Lender or a Share Secured Loan at a Credit Union.
Keep in mind, building credit is a marathon, not a sprint. It involves demonstrating to a potential creditor that you can handle credit responsibly. If you have open, active credit accounts that are being paid on time and pay all of your bills on time every time, apply for credit only when you actually need it and use credit cards sparingly maintaining low credit utilization, then you’re going to earn and maintain great credit scores. It would be impossible for you not to do so. This is the fastest way you can build good, solid credit.
I think you’re an attractive prospect because you don’t have a lot of other cards and are a banking customer
If they don’t give you 10k they might offer a lower limit; it’s not all or nothing
I say give it a shot
If they don’t give you 10k they might offer a lower limit; it’s not all or nothing
I say give it a shot
CU's are one of those things that are rather interesting where if you look at them from the outside they are cruel with their membership wall. Once you are in though they are generally very squishy with fun rules that make it easier than a bank to grow. Most cases the score for approval will be low and the SL's will be more generous. In your case as a member it actually may be as simple as asking the underwriter. I know when I joined the FedEx Employees Credit Association their CC application gets manual review. The reviewer said the minimum score for approval is EQ 570. That is quite low, even Walmart MC will deny before that with TU. I was applying with a score in the 700's and got the max 10k SL as well as the lowest APR.
The caveat is though if you have low income, no matter how good your score is, it wont help you get any products. CU's are just like banks in that they prefer helping their wealthier, or at least more profitable customers. Ime and imho of course