> I realised that most protection Ic need two external transistors, which
> limit the charging and discharging current.

They don't limit it, they completely turn off on overcurrent (usually 
discharge; don't know about charge, don't think so).

You seem to have a misunderstanding about these ICs: they are not there 
to handle normal operation conditions - they are protection devices to 
avoid catastrophic failures (fire, explosions etc).  When they activate, 
the battery may already be damaged.

That may explain the rather high trip voltages - they are meant for some 
Amps, not Milliamps.  Btw, there are two transistors in series, so each 
one takes half the resistance.  And nobody stops you from adding an 
additional resistor.  But low power MOSFETs are easily in the hundreds 
of Milliohm, if not Ohms range.

>> When they activate, the battery may already be damaged.
>
> Does this not depend on how I choose the transistor?
> As I want to protect the battery, I need to choose the transistors so
> that they switch off before too much current can damage the battery.

You can try, but it's not what they are made for.  I.e., the tolerances 
are rather high (DW01: slow trigger: 120-180mV, quick trigger: 1.0-1.7V; 
overcharge limit: 4.3V; etc).  For charging you need an additional 
current limiter anyway and only use the (too high for normal use) 
overcharge protection.

> Is this not the purpose of a battery protection circuit?

No, think of them like fuses or RCDs.  They protect you from an 
exploding battery ;-)

If you want a charger, why don't you take a charger IC?  They know how 
to charge batteries (CC/CV mode), they have a current regulator, they 
have precise references and they will stop at the right time.