We do something similar (actually even more restrictive) and I hate it. Reimbursement turnaround is usually not bad but if I incur sizable interest, I will submit a second request later on to pay that. By policy, that's not supposed to be allowed but I have managed to swing it in a couple of cases by pointing out the cost I incur due to AP not hitting their expected processing times. A few things I would recommend considering:

-The person in charge of finance or accounting at your organization should understand the reality of having employees with tight credit lines in this age. It's not worth putting yourself in a rough spot to avoid potential embarrassment. If it's a small organization, there may be some flexibility on their part. I would just have a direct conversation with whoever acts as the controller to discuss options. You don't need to involve everyone from your manager to the AP clerk, just go directly to someone with oversight of that process. Whoever you speak to just needs to know the situation at hand, not the circumstances that caused the poor credit.

-You can call your CC and ask for temporary limit increases. They should be more willing to accommodate this than a permanent increase.

-If it's a frequent issue about being paid back on time, I would recommend keeping track of how much you're incurring and pointing that out. You may be able to work out a process where you can submit for expedited processing if you can point out the hardship.

-The ideal situation would be to just move to corporate liable cards. They will have a higher limits than employees with otherwise poor credit. It also forces compliance with corporate rates, policies, and lets the company leverage travel volume for discounts. It ties up corporate funds instead of employees' and requires more management oversight but IMO worth it. If you're going to be traveling frequently, it may be worthwhile to have a conversation about having a corporate card issued to you.

-For your situation, a more likely solution may be to have the company issue you a re-loadable gift/debit card. The card is to be used only for travel expenses. You'd use the debit card to pay for charges and then use your personal card to cover incidentals or deposits. It's similar to petty cash but does provide some better oversight or comfort for your org. It's also more fraud proof than having an employee responsible for holding and handing out cash. Obviously you would need one that allows you to pull transaction histories. You would need some "room" on your personal card but any temporary holds would still fall off faster than you are being reimbursed.