All "playing" with LiIon batteries invites flaming meltdown.This is unlikely but possible in this case if due care is not taken (and just possibly even if due care is taken).
If you are not prepared to accept this outcome, however unlikely, then don't play.
It's probably OK, but ...
You need to know if a significant part of the charging circuit is inside the trimmer or if essentially all the charger is in the external power supply. It is probable but not certain that the trimmer simply presents the battery to the charger for charging.
a. Measure charger open circuit voltage.
b. Apply an 8 ohm load and measure charger voltage.
This loads the supply to about full capacity (as I =- VrR = 4.2/8 = 0.6A)
c. Briefly load the charger with a 5 ohm (or 4.7 ohm) load and measure voltage. (a second should be long enough with meter leads preattached to resistor). .
d. All care no responsibility test: Apply a current meter on 1A or better range via a say 2r2 resistor and measure current. This can be a short sharp test just enough to check a stable reading. Well under 1 second. Some few supplies will blow a fuse or have a fatal hissy fit. This is very unlikely but possible. I'd try this test on my personal gear and conclude that the charger had been a bad one before I killed it if it died BUT you may noyt want to risk it.
Open circuit test: The unloaded voltage in a. should be 4.2V nominal and not much more than that absolute worst case. 4.3V actual at room temperature is too high. (The extra 0.1V = 2.3% extra and many meters are only about that accurate so you need to know how good your meter is to make this measurement with certainty).
If the unloaded voltage is above 4.3V the charger is dangerous.
Max load test: The loaded voltage in b. should be in the 4.0 - 4.2V range. Above 4.2V means the charger is very dangerous.
Below 4.0V is not bad per se but means the charger does not meet its spec. No problems.
Flat battery test: The loaded voltage in c. should be about 3V - certainly well below 4V2. This is an indication that the charger is current limited to around the stated level. As R = V/I and 3V / 0.6A = 5 ohm it means that a discharged LiIon cell with about 3V terminal voltage should draw the maximum rated current and reduce the voltage to match. A voltage much above 3V here means that the "charger" is NOT a charger.
d. The current short test is an excellent way of determining if there is current limiting and what level it is at. As noted, a few supplies will only do this once :-(.
ALL CARE NO RESPONSIBILITY BARE BONES CHARGER:
(1) Buy one of the manu USB to LiIon chargers on the market, or
From the above - IF the charger is a USB charger it will -
- Apply close to 4.2V on open circuit, and not more than say about 600 mA when loaded to around 3V. This means that a LiIon cell of low terminal voltage will be charged correctly.
IF the charger meets the above tests it is a complete charger by itself and you can use another USB charger in it's place.
A simple but acceptable charger in most cases is an LDO regulator rated at at least desired Ioutmax and set to 4V2 output at 25C
and fed from USB with a 3R3 resistor. The resistor should be at least 2W rated and preferably 5W.
The LDO will prevent the output voltage from rising dangerously high and the 3R3 resistor stops current exceeding the 600 mA limit when USB Vin is at 5V and battery is at 3V.
ie Imax = V/R = (Vin-Vout)/R = (5-3)/3r3 = 0.6A.
In practice the regulator will also drop some voltage and further resuce the current.
For the above to work the LDO must be well behaved when in dropout - ie drop a minimum of voltage while passing offered current.
The above does NOT limit current supplied if Vbat is dangerously low - which could lead to "vent with flames" situations if the battery is every very deeply discharged. This should not happen in normal use if the appliance shuts off whe Vbat falls too low. This is an utterly essential spec for any LiIon operated equipment but there is no certainty that all equipment does this.
The above charger is overly simple for best performance but is an easily achieved starting point that works OK in most cases.