We are currently in the month of December and have about $10K (all revenue - all expenses) in our LLC's account. We plan to use this to invest on a project early next year. Unfortunately, it is not possible to start and pay for this project by the end of this year. Is there any way we can avoid paying taxes this year?
No, because in reality: it is income, not investment. Regardless of how you turn it.
At the end of your financial year, surplus per definition is income.
I assume you're a cash-basis taxpayer in this case. The answer to your question will be "No," you can't skip paying taxes on your net income for the year. The only way I know is to actually spend that money on something which qualifies as an expense. If you can do that, then you can achieve your goal. Can you pay somebody upfront for the work they will perform in the next year? I advise you consult a tax attorney or CPA for your particular situation (your question unfortunately does not provide enough info for a better guess about the best possible solution).
If you can prepay rent, phone bills, or anything else you can stock up on, it will reduce your income. But on $10,000 how much income tax are you worried about. Some states have a minimum tax anyway so you might as well turn your income into after-tax money if you have to pay a minimum tax. If you're not rich and in a high tax bracket otherwise, the tax won't be a lot on your income anyway. You have to calculate how much tax you're looking at. But in the end, you have to account for it.