Hi, the first ones l think would be a direct replacement for the OEM Oset batteries (check physical size to make sure they fit), probably no better or worse. I suspect the 2nd ones would work, and they would be much lighter. The biggest concern is whether they can deliver enough amperage at peak demand. I'm not sure what the 12.5 Oset draws, but those batteries have a pretty low rating so it'll be iffy. The most straight forward upgrade imo would be a custom battery pack of Lion cells to suit it. Places that make battery packs for e-bikes can usually do one up for an Oset. For a 5yr old on a 12.5 I'd go pretty big. Probably 7s5p or 7s6p with Samsung 40T cells. That would give 20ah. Packs like this is what Boost does which many folks here recommend. I bought one from them, it wouldn't physically fit, and dealing with them was an exercise in frustration and misery and I lost money. That was years ago and they are still in business so presumably they at least some times manage to make packs that work.
A quick summary of battery info is below in case it's helpful. When my son was on a 12.5 we ran a Lithium Ion pack in 6s4p of Panasonic 1850 cells (25.2v, 12ah) from a wreaked Tesla. We focused on small, light, and durable as he was 2 at the time. If setting up for a bigger kid it'd make sense to add some cells for more capacity. On his 20R we have a pack underway that will be 14s6p of Samsung 40T cells (58.5v, 24ah).
Sealed led acid (SLA) [these are what I take out of Osets, ha!]:
Pros: Cheap, simple, rarely catch on fire, easy to charge.
Cons: Very heavy, voltage sags substantially when in heavy use reducing performance and range, physically large.
Lithium Ion cylindrical cells (e.g., 18650 and 21700) [these are what I run in the Osets]
Pros: High energy density, very long life (can re-charge thousands of times), cells can be configured to customize battery pack. Very little voltage sag so performance is quite good. These are what are in Teslas. As lithium batteries go, they are pretty durable and hard to catch on fire. C ratings are relatively low as lithium batteries go, making them ill-suited where huge amperage draws relative to amp hour of the pack is required (e.g., fast RC cars where we pull 300 amps from a 7ah pack). Much faster charge capability than SLA (can charge around 0.5 C safely, with good chargers and some cooling you can push it harder). The cells are physically pretty durable. It's relatively easy to find custom pack builders who will make you a plug and play system with BMS and provide a charger.
Cons: Good ones are expensive, cheap ones are utter crap (never buy from China, they'll be Chinese fakes). You want Panasonic or Samsung. You need a Battery Management System (BMS) or very fancy charger to charge them. Ultimate results are all about the specific cells used and selecting the right one for the application. For most Oset applications Panasonic NCR18650Bs or Samsung 40Ts would be optimal. C rating for discharge and charge will be much less than LiPos. Building a pack from cells requires skill, a spot welder, and pure nickle and copper strips (sandwich them with Nickle on the outside). Compared to SLAs they are more sensitive and if you get something very badly wrong they can cause a fire.
Lithium Polymer (LiPo) packs. [Note, I have lots of experience with running these in big/fast RC cars]
Pros: Cheaper than good Lion packs. It's possible to have huge C ratings (note that most of the times these are hugely inflated so don't believe what the sticker says, especially if you are looking at really low end packs like those from hobbyking). These exist primarily for the RC car market and are available in a variety of pack size and configurations. Buy a few, solder up some jumpers, and you are on your way quickly and easily. Due to huge C rating it's possible to re-charge in <30 min. Due to huge C rating there's almost no voltage sag.
Cons: Difficult to setup a BMS system, so you have to manually keep them happy, balance charge, and be super sure not to over-discharge (that will permanently kill them). Get any of the care and feeding wrong and there is high potential for them to go up in flames. They are often physically not protected, so in your install you have to make sure they aren't prone to physical damage. If damaged, high probability of fire.
Pros: These are typically configured in the same size/shape/voltage as car or motorcycle batteries, so it's an easy drop-in swap. Much lighter than SLAs and much less voltage sag. Physically you usually get the same amp-hour capacity for a given size, so upping Ah with these is difficult. Good ones usually have built-in BMS to protect things and the chemistry is relatively resilient, so fire hazard is low.
Cons: Low power density for their size. You have to pay close attention to the included BMS to make sure the amperage capacity is there. Relatively expensive per Ah.
S = Series
P = Parallel
Ah = Amp Hour
BMS = Battery Management System
Wiring batteries in series means connecting positive to negative and so on. The resulting voltage will be the voltage of the packs/cells added up. The Ah will be that of an individual cell.
Wiring batteries in parallel means connecting positive to positive and so on. The resulting voltage is that of an individual pack/cell. The Ah will be that of the individual packs/cells added up.
So 6s4p for example means 6 cells in series and 4 in parallel. If you do this with 3ah, 4.2v cells (very typical), the resulting voltage is 6 x 4.2v and the resulting Ah is 3 x 4.