As an avel can I make a l’chaim for my recently engaged child?
You may make a l’chaim in your house. You may participate and eat at the l’chaim, mezonos and drinks as is customary, but should not participate a full seuda. You should not participate in dancing or live music.
See Rama Y:D 391:2 that there is more leniency in one’s own house.
See Gesher Hachaim 21:8 with regards to participating in a mitzvah gathering without joining the meal.