Bermuda grass is non toxic-- so fresh or dried, it is okay---Do not feed just the Bermuda-- do include other grasses and 'weeds' as the case may be-- to help ensure a balanced diet.
each plant, in Nature, provides different nutrients, different protein building amino acids, and, as you have seen by comparing timothy and Bermuda-- the fatty acids differ.
An example would be the deep rooted dandelion-- which can pull up nutrients from layers of clay/soil that are beyond the reach of alfalfa or timothy! Plants that grow in wet areas will have different mineral levels than those that grow in drier areas- even if it is the same plant! WHere a pkant gorws, ow it is managed during it's life cycles, and even how and when it is harvested, all affect the nutrient levels of the end product--whether fed fresh or dried, as a 'hay'