How can I apply for naturalization after my I-829 was denied?

You can try, and the case may just be held until I-829 approval.

Legally, you are still a permanent resident, so technically, you can apply but current policy is not to approve unless the I-829 is approved. May have to challenge this in court.

There is no appeal of an I-829 denial. I assume, you mean you will present the case again in immigration proceedings before a judge. You can receive I-551 stamps to give evidence of your CPR status until final resolution of your case in immigration court. You could apply for naturalization but it would not be adjudicated until a final decision is made on your I-829.

Unfortunately, no, the permanent residency must be granted before or at the same time as the naturalization.

What does your lawyer say?

You cannot apply for naturalization unless your I-829 is approved. Therefore, you must wait until your appeal succeeds which will serve as basis for naturalization.

You must wait until the appeal is granted. Your conditional status has not been changed to Lawful Permanent Resident.

The only way for you to receive a permanent green card through your EB-5 case is for your I-829 to be approved, either through appeal or by an immigration judge in removal proceedings. You can apply for naturalization since you meet the requirements, but your naturalization application cannot be approved unless and until your I-829 petition is approved, and you receive your permanent green card.