Battery protection circuit
Note how the seller lists "Nominal Capacity:3000mAh" in the specifications while this clearly is a 300 mAh battery. That 3000mAh is a typo for sure, however I would not rely on anything listed in the offers from most Chinese sellers as they often have no clue what they sell. There are exceptions of course.

From the photo it can be seen that there is some (yellow) kapton tape near the connections and I think I also see an SMD resistor under the tape. That indicates that a protection circuit might be present already. However, there is no guarantee that you will get what is shown in the picture.

If making the protection PCB yourself is not needed for your project you could consider buying a ready made protection module. Or you could buy a battery which is guaranteed to have build-in protection. There are many generic batteries intended for old fashioned (not smartphones) Nokia phones which are cheap and have the protection. These might be larger in size than this 300 mAh battery though.

With such a small battery I would not exceed the 0.5 C discharge current, that is 150 mA. Drawing 4 to 6 A from such a small battery is asking for trouble.

I do not think the TI BQ297x is good choice for such a small battery, it might be better to choose a protection IC designed for small batteries. Such an IC might also have everything built-in so no need for external MOSFETs.