As budgets tighten, what are ways that I can position my product as a priority and worth keeping within the budget?

Make yourself relevant. Always tie your solution to strategic company goals.

If it is all about productivity and cost savings, show how your solution can help achieving that - the more precise your numbers and the business case reflects their business, the better.

If it is all about increasing revenue, show how you can accelerate time to market or increase conversion rate. Proof it with references, the closer to your prospect's business the better.

Additionally indicate the costs of doing nothing when the prospect is hesitant to invest the tight budget in your product. What does they loose if they do not start today but in a year?