If the offers are similar both in AD and Bahrain, you will get more bang for the buck in Bahrain and can definitely save a lot more there because of the lower cost of living.

However, the downside is that there is not much in terms of recreation and activities for your child which may be something to think about.

Accommodation in AD is ridiculously expensive. If you are being provided accommodation rather than an allowance, go for AD as there is certainly a lot more life in UAE than anywhere else in the Middle East