No that quote is too high, especially the 300 side of the quote. You are paying a company, not the installer who gets less than half of that. You should seek out an installer and be able to get that done for ~100, which is reasonable.
Doing the job yourself will be a little more difficult than you might want to get involved in.
First you need to get the knee kicker, then tack strip, some reinforcement nails, something for the room transition (metal, wood, etc.).
Second you are dealing with Berber carpet, which will be difficult to stretch and cut. The berber carpet is stiffer than regular cut pile carpet, so any bubbles/wrinkles will be harder to kick out. The continuous fiber strand will be prone to snagging if not fully cut, pulling across the room and leaving a gap. Plus the chance of leaving cut marks on newly finished baseboards.
But this can be done if competent enough.
When cutting the tack strip, you can use tin snips. It won't damage them for the few cuts you will make. Lay the tack strip against the wall with a distance of 1/2" (approximately the width of your finger). Make sure the tacks are leaning towards the wall, so as to catch the backing of the carpet when stretched over it. The gap you left is where the carpet will be tuck into when trimmed.
If nailing into concrete, make sure you use smooth deliberate hammer strokes so as not to have nail deflection. If nailing into anything harder than concrete, use a 5/16" masonary drill bit and use 3/4"stub nails to secure the tack strip down. Use reinforcement nails where ever necessary.
Before laying the carpet down, make sure the padding is up to but not on the tack strip. Trim if necessary.
When stretching the carpet with the knee kicker, try not to drag your knee or you'll end up with a really sore knee/carpet burn. After each knee kick you will rub/press the carpet onto the tack strip, using a hammer that is clean, to secure it. Keeping the knee kicker perpendicular to the wall, when kicking.
For trimming the carpet, you will want to have a very sharp knife and take your time. For this small of a job you can use a utility knife. Fold the carpet back from the wall without pulling the carpet off the tack strip. Without pressing down too hard on your knife, make a cut a 1/2“ from the fold along the wall. If you have trouble cutting the Berber strands, you can cut them with a scissor. When done with a section, tuck it in with a spackling/paint tool (a 3-n-1 will work as well).
Just take your time if you do this on your own.
Best Answer
In a normal room-sized install, both tools are used. The kicker for the first edge and the stretcher for the rest. Kickers are also used in corners where the stretcher is awkward to use. On stairs, just a kicker is used.
A kicker is ideally only used to attach the carpet to the tack strip.
A stretcher is used to attach the unanchored sides. You anchor it at an attached side and stretch the carpet to the opposite side.
A pro would use a stretcher as much as possible. A carpet that has been kicked with a reasonable amount of effort will pull up at the edges from any typical abuse.
A small space like 2x3 would have only used a kicker - which is why they invented the mini-stretcher (like this one). They're not cheap for a single job and I haven't seen one for rent, but it's worth a few calls around.
That's not to say you can't just use a kicker, and I probably would if I didn't have access to a mini-stretcher. You can do a 90% job in a small space with a kicker, but you'll definitely work up a sweat.
FYI: For larger jobs, even with maximum effort, the smaller head of the kicker will lead towards ripping the carpet if you try to get the same amount of stretch as the power stretcher will give you with no effort.