A Honda Odyssey is a unibody vehicle. Conventional wisdom is that body-on-frame vehicles are best for towing, and for heavy trailers that is generally true. However modern unibody vehicles are much improved and make great tow vehicles for smaller trailers.
With an Odyssey I think you could step up to a 15' or 16' if you wanted and could find one, provided it's not too heavily optioned or loaded. A 15'
Trillium with front bunk sleeps 4 and has a larger main bed (54" vs. a rather tight 45" on a standard
Scamp 13'). It will run around 2200 pounds loaded. A
Scamp 16' layout 4 gives you a small bath and layout 7 (rare) gives more space and an extra bed when kids outgrow the upper bunk. Either can be kept under 2500 pounds loaded.
On the other hand, I can testify that a 13' bunk model makes a great starter trailer. It's all a small family really needs, and it's easily re-saleable if you decide you want a larger trailer later.
No WDH is needed, but an anti-sway bar is a simple and inexpensive add-on safety device. Any yes, you want trailer
brakes and a controller for anything over 1000 pounds (per your owner's manual), which includes all 13' trailers. Not all have them.