Since you do have coupon pages (in general: pages), you should add them to your sitemap. There is an option to indicate how often your page gets updated via the
<changefreq> but keep in mind it is often not taken into account by Google.
To avoid returning 404 errors to googlebot consider implementing a way to return 410 status instead (a resource used to exist but now is
gone). This way you will signal that a page is gone in a more meaningful way instead of returning a 404 error.
When it comes to crawling budget, this case will not affect it. Crawling budget is affected by how fast your server responds, the SEO of your page, the loading time of your page etc. When it comes to non technical aspects, crawling budget is also affected by your site's popularity, your content quality, the number of visitors etc.
Your case, a temporary page that soon-ish ceases to exist is not one of the factors that affect crawling budget. You can read more about it in this official google webmaster post.