Hi,
I think the first makes no sense, The second and third are both natural. I think the third suggests a more personal commitment.
While keeping in touch seems to suggest a mutual effort it doesn't always mean that. If you say, "Let's keep in touch, " it could mean "I'd like you to call me and I'd like you to say it's okay for me to call you." But if you say, "Is it okay [for me] to keep in touch with you?" you're usually asking for permission to initiate the contact. If you want the person to call you, you'd probably say, "Please give me a call", unless you were afraid of being too forward, in which case "Let's keep in touch" would give either of you an out. I'm really not sure what you'd like to say in the first example. It sounds to me like, "Is it okay for you to call me?" which makes no sense. Perhaps, "Would it be okay if we kept in touch?" would work.
I suspect cultural differences would be more important here than differences in language. Opinions to the contrary are certainly in order.
Regards, -A.
P.S. Your post was listed as "unanswered." When I replied, two popped up from five hours ago. Sorry. I think "Please keep in touch" is great!