Jobs I got are; BST, mnk, pup, blu, rdm, war, drg, cor, rng, thf, whm, pld, nin, and brd.
I'm really looking something for nin, mnk, pup, thf, bst. Dunno if these as will outdo rampage or stringing pummel for pup
First, 5/5 a merit WS or don't bother. At lower levels, the existing WS will beat or equal the merit WS. So you only have room to do 3 full 5/5 merit WS. If you're focused on the five jobs you listed, I'd say 5/5 dagger for sure and then pick the other two based on the jobs you plan to use more often out of NIN, BST, and outside-Abyssea MNK.
1) Get Exenterator unless you plan to get an empy dagger. It's really good for THF (plus you can get some use out of it on your other jobs, particularly as a good COR melee WS)
2) Blade: Shun - clearly the best katana WS if you aren't going to get Blade: Hi. Downside is that you need to use 5/5 merits for a WS that's only usable on one job. I'd personally say go for it given your priorities, and if you end up doing a Kannagi (one of the easier Empyrean weapons) you can switch to fulltime Blade: Hi and drop your Shun merits to free them up for a different merit WS.
3) Shijin Spiral... I also have MNK and PUP as a couple of my key concerns (for me it's PUP MNK DNC THF NIN COR). For PUP Shijin is not really that big of a deal. Stringing Pummel is superior to Shijin in Abyssea, and generally no worse than a sidegrade outside. However, Shijin is the best non-Empy/Relic WS for MNK outside of Abyssea. Your call on how often you use non-Abyssea MNK.
4) Axe... I really don't know much from personal experience. However, given that your pets will do a lot of your fighting for you, might be a consideration for sticking with Rampage from the BST.