Using LiPo cells without a protection circuit
The protection circuit protects the LiPo from many dangerous conditions. Overvoltage can be ignored here, as you are not going to charge the cells from your circuit. Undervoltage (i.e., overdischarging the battery) can be fatal to the cell, although not immediately dangerous (the big issues might arrive later, when charging). The other protections that are common are overcurrent/short-circuit, which is a very important one, as a shorted LiPo can potentially explode, and overtemperature, which is mostly useful while charging, or while discharging at very high rates.

For your particular application, if you take care all the time of the battery voltage (making sure it does not overdischarge), and only use a proper charger to charge it, I don't see a major problem in running your circuit for test purposes without the protection. However, make sure you add a fuse between the battery and the power input of your circuit, to prevent dangerous conditions in case of overcurrent or short-circuit. This is very important. LiPo batteries can be very dangerous. If you have a LiPo protection bag, use it, as long as you're testing without the protection: better be a bit too careful than not enough.

Concerning a hand-solderable protection IC: I recently used a DS2777 (for a 2-cell system), it works pretty well, and even if it is a SMD part, it can still be soldered by hand, although it's not simple. But is possibly a bit "overkill" for your application, as it requires two external MOSFETs, and is not only a protection circuit but also a battery monitor system that can estimate remaining charge, read battery parameters (voltage, current, accumulated power, etc.) over I2C, etc.