Is it disliked to shake hands and say "haram" immediately after the two salutations, and is that permissible, or must the prayer be concluded first?

There is no basis for shaking hands after concluding the prayer (salam). Our preferred view is that it is disliked (makruh). Sheikh al-Islam Ibn Taymiyyah was asked about shaking hands immediately after prayer: Is it a Sunnah or not? He replied: "It is not a Sunnah; rather, it is an innovation (bid'ah)." Similarly, making the phrases "May Allah accept [your prayer]" (Taqabbal Allah) or "A blessed sanctuary" (Haraman) a constant habit after prayer is disliked and an innovation, because it gives the impression that it is a Sunnah. However, if someone extends their hand to shake yours, do not refuse it; rather, shake their hand and gently explain the ruling concerning this act. Likewise, if they say to you, "Taqabbal Allah" or "Haraman," thank them, then explain the ruling to them.