You are allowed to deduct the ordinary and necessary expenses that you incur in connection with your LLC business.
Internet access is certainly an ordinary and necessary expense.
1. Your allocation of 3/7 of your monthly internet bill is a reasonable method to allocate the expense since you are using the internet 3 days of every week (3 divided by 7 multiplied by your $100 monthly bill equals $42.86).
2. Pay for your internet with your business credit card. A single member LLC is a "disregarded entity" for IRS purposes so you will file the business on Schedule C. There is no need to implement a reimbursement process; just adjust your total expense downward for internet access when entering your expenses for the business on Schedule C.
I do like the idea that you are working to make your business accounting simpler by getting all expenses on the card; that is why I suggest that you pay for the internet on the card, that way you will not forget to enter internet expense when you do your tax return.
