Can I sue personally instead of through my LLC to get the higher small claims limit?

Only in the narrow situation where the underlying contract was genuinely formed with you as an individual person, rather than with your business entity. If your invoice, the underlying contract, and the bank account that received payments all identify your LLC as the contracting party, attempting to sue in your own personal name instead invites a standing challenge, a legal argument that you personally aren't the proper party to bring this claim, and that challenge will generally succeed against you.

Sole proprietors are a genuine exception to this problem, since a sole proprietorship has no separate legal existence apart from the individual person running it. If you operate as a sole proprietor rather than through an LLC or corporation, you get access to the full $12,500 individual limit rather than the lower entity cap.