Many Panamanians take an afternoon siesta
"Many" indicates by no means "all" so it seems whether the client quietly disappears until siesta finishes or not will depend on the client. You may have no choice but to "play it by ear".
As you quote, the siesta tradition is connected with the heat (it is also connected with a large meal at lunchtime). But you also mention the residence is likely to have air con so heat no longer the issue, just custom and habit. It is certainly the case in Spain for example that it is a dying custom. Taking a hunk of time off during the day just extends the interval between starting work and ending work and in some cases will mean four commuting journeys per day instead of two. The trend worldwide is to favour more time in the evening at the expense of losing some of a break at lunchtime and eating less for lunch (eg fewer subsidised canteens and more take-away sandwiches). A switch away from agricultural to office work (dawn start and manual labour to office hours and sedentary or light work) probably means we are less tired by "the sixth hour" anyway, and less hungry.
The same source also has:
Office opening hours vary but generally businesses are open from 8am to 12pm, then 2pm to 5pm or 6pm on weekdays, and 9am to 12pm on Saturdays.
That is a 2 hour break in the middle of the day which may in practice leave only about ½hr for a nap. In Spain with some mid-day breaks of 3 hours any nap could be three times as long as that. In any case, from around November to April the heat is not really getting towards intolerable in Panama, even out in the open, so for half the year heat is not really much excuse for a siesta.
Should I expect to be offered a spare bedroom?
Basically, I have no idea (!) but I would not expect that. A siesta can mean getting undressed and going to bed but I think more often is just a kip on a couch (fully dressed) so a bedroom unnecessary. I take it your host is well-educated so will surely not expect you to require a siesta to continue with what is habitual for you.
What can I expect as siesta time approaches?
A general slowing down of pace (shops and businesses may be shut and fewer people on the streets) but if indoors probably little discernible difference.
Should I return to my hotel and wait?
I doubt it. That just seems unnecessary time/cost wasted in travel. Even if your host does disappear for an hour or so I would expect that you would be trusted on your own and would find something worthwhile to occupy yourself (email, a swim, ... ?).
But the real question is:
What's the etiquette for a guest who's on business in Panama at a private residence when siesta time approaches?
Etiquette is consideration for others but I think exactly how is unanswerable because your host could about as easily prefer to "work through" as be a 'traditionalist' who demands a long siesta. I think you should just take your cue from your host. I expect her/him to delight in your company and be keen to have you around as long as possible but, since a client, might also want to be shot of you as soon as possible (to save on your fees!).