Attacking this from a different angle have you considered a smartwatch?
There are really two major platforms (if you rule out Apple Watch based on price) which are Pebble and Android Wear. I picked up a Huawei Watch recently (came free with the Huawei P9 I brought the mrs) and am loving it. It's the model that google are pushing so one of two that gets the preview update of Android Wear 2.0. It doesn't do the temp and baramometer but does lots of other cool stuff, shows you your notifications, remote shutter for your phone camera, can store music on it so you can listen without your phone, lets you control your music and it even tells the time!
The battery lasts two days, it has a saphire screen and unlike the Apple Watch the screen stays on all the time and it's actually round like a watch should be.
Because they are marketing the P9 quite hard and it comes with the watch there are quite a few on trade me selling for about $300.
Pebble is the other major brand. They have their own eco system and app store and traditionally have used an eink screen which means the battery lasts a lot longer.
Both of these brands can be used with either iOs or Android.
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