I'd say March is slightly more pleasant than October. Less rains, less moisture, less sweating.
Re: the pools. Last time I stayed at Amanbagh and Aman-i-khas, the pools were uncomfortably cold. Though the Amanbagh pool has some temperature control, it is woefully inadequate for any weather except summer proper. If you are at either of these properties in March or October, I'd expect that you'll experience the same thing unless they've fixed this.
If you haven't stayed at the Oberoi vilas properties or the Taj palaces, I'd strongly suggest adding these if at all you can. In Ranthambore, particularly, I thought the Oberoi Vanyavilas was decisively better than Aman-i-khas. Amanbagh, on the other hand, is fabulous.