OsakaWino

If the ryokan you're staying at is in an osen-gai (an onsen street or neighborhood) like you've seen before and like in your photo, i.e, an area with a number of ryokan and/or onsen hotels with numerous souvenir shops etc. where people walk around in yukata, your ryokan will most likely provide a yukata that is acceptable to be worn outside. But when staying at a ryokan that is not in an onsen-gai, people do not go outside in yukata.

I did not see anything like that in the immediate vicinity of the HR Hakone. I'm sure you know that the surrounding neighborhood is mostly company retreats, private vacation homes, vacation condos, etc. Not even very pleasant to walk around in street clothes, nevermind a yukata.

If you do stay in a area where you are comfortable walking outside in a yukata, you could certainly take a taxi over to the Hyatt for a drink while wearing your yukata. I cannot imagine that the Hyatt staff will think anything of it; my impression was that they have seen a wide range of somewhat eccentric foreigners.

We have always considered September a great time to travel in Japan because it is in between the traditional Japanese travel seasons of August (especially around obon) and October.