How can Arrhenius bases also be Brønsted-Lowry bases?
Sure it can.  You're just incorrect about the products when $\ce{NaOH}$ accepts an $\ce{H+}$ in aqueous solution.  The result is $\ce{H2O(l) + Na+(aq)}$ instead of $\ce{HNaOH+(aq)}$.  That is, the $\ce{NaOH}$ doesn't stay together.

Now mind you, most people would point out to you that the $\ce{NaOH}$ doesn't stay together as soon as it's dissolved, meaning $\ce{NaOH(s) -&gt; Na+(aq) + OH-(aq)}$ happens right away, and then as soon as you have some $\ce{H+(aq)}$ around it reacts only with the $\ce{OH-(aq)}$ and the $\ce{Na+(aq)}$ doesn't get involved, probably isn't even anywhere near.  But this doesn't change the main point, which is that if you add $\ce{NaOH}$ to water, you have a solution that can react with $\ce{H+(aq)}$, which means you added a Brønsted-Lowry base.