Yes, you absolutely should rescue the climber when the situation allows.
The reason that matters most is suspension trauma: Prolonged motionless hanging in a harness can lead to loss of consciousness and eventually even death. Of course you say the victim is conscious, so he might be able to move or even install a foot-loop to transition weight to his legs. You also state that he can't climb out, so it is likely he is injured so he might not be able to do this. In any case, even if he is not in immediate danger for suspension trauma, free hanging in a crevasse is not a comfortable position to be in (personal experience). So assess the situation together with the climber in the crevasse and take a decision.
I am not saying you need to get him out in every situation, but if the situation permits it, you should. So as usual contact the victim to evaluate his condition and position. If he is unhurt, able to get weight off the harness, but for some reason can't ascend and rescue is hard to do due to an immense snow lip at the crevasse border - by all means wait for the professional rescuers. They have material like tripods to solve such a situation.
There might even be a legal argument about denial of assistance if the victim comes to harm due to the prolonged stay in the crevasse and it can be proven, that you were able to help but didn't. However I doubt this can happen unless you are a professional yourself. Still this is also morally relevant: Do you want to be in the situation where you need to ask yourself the question: What if?
As I raised legalities I assume the following argument comes up: But what if the climber is injured during rescue?
The premise is that the climber is conscious. So you can communicate during the entire rescue and you should. Then the risk is minimal.
I specifically didn't go into details of the rescue, as that was specifically not what the question asked: They are experienced in crevasse rescue. So when I say you should help when applicable, I obviously mean to the extent that makes sense given the situation (I specifically said so) and under the number one principle of any rescue: Keep yourself safe.