No. The peer reviewers who score your application don't use the budget to assess scientific merit. It is an “Additional Review Consideration” with no effect on an application’s overall impact score. That will remain the case under the Simplified Peer Review Framework that takes effect early next year.
Since feasibility is evaluated as an aspect of “Approach” under “Rigor and Feasibility” of the new Simplified Peer Review Framework, if you drastically underbudget for how much your research would normally cost, reviewers will likely question whether you really know what you’re doing. Although the budget is not a scorable element, a request that is exceedingly low could negatively affect your review outcome if reviewers then question your project’s feasibility, which is a scorable concern.
Simply ask for what you need and justify it.