I know there are several electric formula teams running their cars with these Energus modules, so I guess it is worth the money. I am thinking of these modules myself, following the company for some time now, but did not yet commit to buy.
If you are novice with lithium cells, it is better to buy than to do yourself, it will be cheaper in the end. Just make sure yiu use a good BMS, that yiu can trus your home with. Energus offers that as well, if your are not going above 60V nominal.
I managed to obtain 6 x 18650 Batteries from an old laptop.
This is your first problem. Those old batteries are probably tired and will struggle to supply the required current. Individual cells may have different internal resistances and capacities, so balancing is advised.
Solution A - Use only a 1S3P (or more in parallel) Pack instead and
use a TP4056-based USB 5V Charger.
Bad idea. The battery will charge very slowly, and the booster will waste power. The pack and wiring will have to handle 14A+ discharge current.
Solution B (BMS and '12.6V' charger)
If the BMS includes balancing then it should work, provided the '12.6V' charger is designed for 3.7V Lithium cells. Without balancing, some cells could reach peak voltage before others and then the BMS would terminate the charge early, resulting in a partially charged, out of balance battery.
The BMS won't cut on discharge until at least one cell has dropped to a dangerously low voltage. After a few cycles the cells will start dying. To protect the battery you should install an alarm or cutoff that doesn't let any cell go below 3.2V.
Solution C - Individually Protect Each CELL with a 1S BMS, AND use a
3S BMS
Overkill, but perhaps (depending on the balancers) not enough! Many balancers work on the principle of bypassing charging current when the cell reaches peak voltage (4.2V). The problem with this method is that if the balancer can't bypass all the current then the cell will continue to be overcharged (until the protection circuit kicks in).
Solution D - The Proper Balanced Method , which would need a use a of
bulky balance charger
Again, how well this will work depends on the particular charger. Some contain 3 isolated circuits that charge each cell individually. This is the most reliable method of balance charging, but the control panel has to communicate with all 3 chargers while maintaining isolation, so it is mostly used in simple low-end chargers that may be unreliable.
More sophisticated balancing chargers have an LCD screen and are fully programmable. Their balancers usually work throughout the charge cycle so the cells start to become balanced before reaching peak voltage, but most of them have relatively weak balancers. The main advantage is that the LCD screen shows you the cell voltages, so you can cut the charge rate down to help balance the pack if necessary. The display also shows how much charge is put in, so you can gauge the health of the pack.
A good balance charger may be bulkier, but will be more powerful and gives you much more control and flexibility. Many can also do Nicad/NiMH, LiFPO4 and Lead acid batteries. One charger may be all you need to charge many different devices.
Best Answer
You certainly can have some fun doing this project. Dead Laptop batteries still might have few fairly good cells to harvest. Keep in mind however that the internal cells don't have any protection, so be careful not to short anything at a risk of "rapid venting with fire".
There are some hackers who do this kind of projects and offer their advice, like this one. In most cases take their advice at your own risk, since usually their methods are not backed up by any test nor reliability/safety research. For example, they would boldly connect 18650 cells in parallel in large numbers without any reservations, which doesn't make them overly reliable and long lasting.
For a overview of some challenges in the area, I just found this nice article, which I highly recommend for a start.
Finally, I am afraid you can't just "shell out money for real batteries" and expect you EV bike to fly. One battery doesn't make a EV. You would need a full EV solution for power management, braking recuperation, etc, all sorts of fine engineering.