You can definitely just use bleach if you want but if you do, you'll have to add some CYA separately which would be fine.

In your small pool, each quart of 6% bleach will add about 2.3ppm of chlorine. Until you get some CYA in there, the chlorine will disappear fast especially on a hot sunny day. I'd add a couple quarts and see if that will give you enough to make it through the day without it all disappearing by evening.

If you decide to go this route (bleach) you'll want to add some CYA to an old sock and hang it in front of a return jet. 2 lbs. of it would give you a reading of about 40ppm which would be good. If you give the sock a squeeze every once in awhile, it will dissolve faster. But, if you do add this CYA separately, you won't want to use the trichlor pucks or dichlor shock powder since they both have CYA in them. When you buy it, it may be labeled as stabilizer or conditioner. Check the label ingredients. You want cyanuric or isocyanuric acid.