Not at all. You are obviously right that at first glance Charriot seems better. It just simply turns out that the Proc is simply good enough to make it worth nevertheless. Many (if not most) caster builds utilize Bat for its sustain. For some builds it is their main form of sustain acually. Basically it is too strong to leave even without synergy. (Btw it also gives 5 affinity of useful colors)
Btw forgot to say: Tempest is not worth the investment and is rarely ever used.
For melee/caster hybrids the closest choice to your original idea is Savagery/Stormtotem Vindicator which has been one of the stronger builds in GD for a long time ya1’s stormreaver vindicator.
A less flashy choice but probably one of the strongest builds out there right now is Dark One’s (Vitality) Ritualist Dmt’s Ritualist. Similar builds are available for Cabalist and Conjurer.
A personal favourite of mine is RoH/Melee Infiltrator, flashy DW action. Strong but sometimes glassy Ya1’s Infiltrator
Shieldbraker might be a choice although usually not 2 handed. Forcewave builds aswell although they are technically not melee.
The strongest choice staying with warder is probably Ultos Primal Strike Warder.
Just have a look through the build compendiums for inspiration. Most builds posted there are very well crafted and will crush Ultimate even if you miss some parts of the build.
That being said I wanna clearify that your build concept is totally fine for clearing MC content and nemesis aswell if well crafted. It will just not synergize just as well as some of the top builds out there. Thats the beauty of GD. As long as a build it is thought through almost any build can succeed to a good extend.