In Arizona, you can apply for just a bonus point. There is a drawback however. You must apply by mail only, no online application process anymore. And, you must also apply for (buy) an AZ hunting license. A general non-resident hunting license is 151.25. The "Bonus Point Only" fee is 7.50 per species you want a point for. If you're only putting in for a bonus point for a single species, then you'll send in your application plus fees for license and bonus point, = $158.75. If you want to apply for multiple species, you'll have to include the fees for each application. You'll only be issued one hunting license, and the fees for the other licenses will be refunded to you. But, either way, you will have one hunting license bought and paid for, and that's each year you apply.
So for example, if you apply for the bonus point for elk, you'll send the $158.75 in time for the elk draw deadline of the 2nd Tuesday in Feb. You will get your bonus pt for elk, AND be issued an AZ non-resident hunting license for the year.
Now, for the deer draw, you will again apply for a bonus pt before the deadline of the 2nd Tuesday in June. BUT, you already have your license (was issued to you in the elk draw), so you'll just include your license number in the application form for the deer bonus pt, and you'll just send the $7.50 fee for the bonus pt.
I hope that answered your questions, as far as AZ is concerned. I can't help you with any of the other western states, so maybe some other state hunters will chime in...