If the power supply is regulated to a fixed current, is there an increased risk of over-charging? Should I use a timer?
The way a typical bench power supply works is you have a knob for voltage and a knob for current. Typically, you'd turn the current limit up high enough that it doesn't come into play, set a voltage and treat it as a voltage source (think thing-that-behaves-like-a-battery), supplying a regulated 5V or whatever for your circuit.
Alternatively, you can flip this on its head, turning the voltage right up, then (probably with the output terminals shorted, so you see something on the current meter, unless it's a funky digitally-controlled one) twiddling the current knob to set some value. Then it behaves like a current source, with the voltage rising to whatever value is needed to push the desired current through the load. Sort of thing you might want to do to heat up some nichrome wire to cut foam, or drive some sort of electro-chemical reaction.
For battery charging, you'd do a combination of both. Starting with the output shorted, you'd turn the voltage right up and set some desired current. Then you'd disconnect the output, and twiddle the voltage down to whatever is the safe limit for the battery. Now when you connect the battery, the voltage will rise as high as needed to push your desired charging current, but only up to the point where the voltage limit is reached, at which point the current will start to drop. Effectively, it acts as a constant-current source, then switches to being a constant-voltage source as the battery voltage rises. Which is about 95% of what a battery charger does (the rest being things like temperature monitoring, cell balancing, timeouts and clever cycling modes to rejuvenate the cells).
A timer's not a bad idea, because there are battery failure modes where it keeps taking power, but does unhelpful things with it like boiling the electrolyte or generating hydrogen. See also: Explosion Containment Pie Dish™. I wouldn't charge a lithium-ion battery like this unsupervised, at least not unless you've done it several times before without issue. Also note the hazard of inviting twiddly knobs that can cause Bad Things to happen if there are other people around (especially those prone to unthinking Tidying Up or who don't respect "Nicht gefingerpoken!" signs).