Personally, since the primary road name for addresses is SR-405, I would put that in as Primary, and add SR-44 as an alternate name.

I'll just go ahead and drop this in here in case you haven't seen it yet, and for future readers. Though it seems like you got your answer already.
Concurrent names

Another factor to consider can also be which route is given preference when they are signed together (on top, or left).

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There are places where we have included both names in slash notation in the primary street name field, but it has been limited to a by-exception basis, where coinciding roads are better locally known by both numbers equally together than either number individually. Also in order to get it done like that, it had to be proposed and agreed to by all area AMs, SMs, and possibly even the RC if disposition among the community was too close to call.

Typically you want to use the highway number in the primary street name that does not involve a turn instruction when they split.

PhantomSoul wrote:There are places where we have included both names in slash notation in the primary street name field, but it has been limited to a by-exception basis, where coinciding roads are better locally known by both numbers equally together than either number individually. Also in order to get it done like that, it had to be proposed and agreed to by all area AMs, SMs, and possibly even the RC if disposition among the community was too close to call.

I'm not sure how these roads are known in the area. So the default option is to use one SR as the primary and the other as the alt and make an educated guess as to which is which?

PhantomSoul wrote:Typically you want to use the highway number in the primary street name that does not involve a turn instruction when they split.

That makes sense and gives me a good idea which to use. Thanks!

None of the Concurrent Names situations really fit here. These are two routes that pass through towns, neither one has a local road name (the route is dual signed with the SR numbers), there are no exits to help choose primary, and the mileage numbering did not yield an answer.

The two things that did help were PhantomSoul's suggestion about turn instructions when they split and also house numbers which the GIS maps showed were assigned to one SR. Both of these gave me the same result for the Primary choice (SR-405 in this case).

Just passing this along as some extra information.

PesachZ wrote:Another factor to consider can also be which route is given preference when they are signed together (on top, or left).

Good point. In this case, SR-405 is signed on top of SR-44, which confirms primary choice.