The Lance.
1. 30% passive Elemental is no joke.
2. PF is useless if you don't Full Burst and Lance gives you a far better chance of reaching 100 Burst. You either attack twice or better yet, use Shingen's Encourage Inspiration for an instant Full Burst.
3. The Bow gives you 500% (up from 200%), so you're hard-capped for the Burst modifier instantly. Which is a good thing, but Exceed weapons are thus half-wasted, as their burst modifier no longer applies (only the cap increase). Same with Burst Accessories, but you don't want Burst Accessories to begin with.
The Bow can be useful, but what is required from you:
1. P2W Eidolon, when using double P2W Eidolons the loss of 30% Elemental doesn't hurt that much.
2. Some way for your Hime to guarantee Shingen to reach 100 Burst again. If instantly, great. For example Shingen's first skill + Encourage Inspiration + 10/20 Burst from a Hime. Remember that reaching -50% Def is very important, so you can't run Encourage Inspiration if your team doesn't cover A or B debuff.
3. While not a requirement, being a massive whale is recommended. Having a out-of-this-world Hime FLB Grid with the right Phantom weapon will let you Burst for your cap despite the lack of 30% Elemental. Furthermore, such Grids typically struggle for HP, so 30% from the Bow would be very welcome.
As you can see, just about everyone is just going to tell you to just use the Lance.Yes. If you're using Shingen's skillset, you'll also get the Energy Release boosts.
If you're using Shingen with Yukimura's skillset, you don't get the Energy Release boosts, I believe.Better isn't exactly right. Elemental and Assault are equally good for damage.
However, this isn't about Elemental or Assault. This is about Elemental or HP. The Axe is offensive - and has the better effect to boot - so you get the Axe, no questions asked.