There will be no significant difference for you as a cross training exercise if you choose recumbent vs upright. The problem you will have is something called "specificity of training," which means, you guessed it, if you want to ride a recumbent bike very well, train on a recumbent bike; if you want to hike well, hike.
In your specific case, you basically need to maintain some level of cardiovascular fitness, which is why your doc suggested a bike or something. It will not really do much for you to keep your hiking muscles in shape - you'll just have to hike to do that.
1) if you have access to a pool, that might be the closest you can get to mimicking hiking if you want to specifically work hiking muscles. Don't swim, per se, just walk, or run, etc. Basically hike in the pool.
2) if you really have plantar fasciitis, it really shouldn't be this hard to treat. Most cases, if handled properly, clear up pretty well within a month or so. Granted, you may have some mild pain leftover, but honestly nothing that should have to limit you. Have you tried the toe splints at night yet? (they hold your foot at a 90 degree angle while you sleep...helps keep the plantar fascia from tightening up overnight.) If you are still having trouble, find yourself a GOOD PT. I cannot stress enough...no PT should waste your time with ultrasound or electrical stimulation or iontophoresis or any of those types of modalities. The PT should be looking at the biomechanics of your foot and entire leg and addressing that - anything else is just placebo and ripping you off. We all know it, but so many clinics just keep doing it because that's what they always do...and the patients like it because it kinda feels good.
But in the meantime, don't fret about upright vs recumbent vs rowing...do what you will actually do and at least somewhat enjoy. Just work big muscles until you sweat and get your heart rate up.
Feel free to PM with specific questions if you want...