Take the electronics out of your existing cheap charger and just use it as a battery connector / holder, wired to the new charger!
Note that you need to connect both the main battery terminals from the charger and (possibly) the correct wires from the multi-pin balance connector for a single cell, to the same two terminals in the battery connector.
I have the Turnigy version; they are a very good charger and have a "storage mode" setting that automatically charges
or discharges the cell to an ideal level for storage. They are normally quite expensive, but just occasionally appear on ebay for less than £15
Note that they do not normally come with a power adapter - you need a power source as well; 12V should be OK for one or two cell batteries.
You have to set the charging current. It may default to two amps or more, for large batteries. Set it down below 500mA or even to eg. 200mA to avoid overheating the small batteries you will be using it with.
These are typical of the balance port connector cables; drone battery packs often have the cable built in, so it does not come with the charger.