no you'll be fine. I do it all the time and i've never had issues finishing a session due to being too full. i do IF and on occasion am eating as much as 2000cals before lifting. typically 3 plates: salad, veggies and something hot (meat). if i'm short on protein i may supp with a protein heavy dessert (protein fluff, hm protein ice cream, yoghurt, etc)
as i see it, you're tossing up between 4 options:
a) lifting then eating
b) eating then lifting
c) eating, not lifting
d) lifting, not eating
option a) is ideal and b) not far behind (the difference would be negligible imho). you say option a) is not an option due to family dinner, d) is not an option due to family dinner and c) is a rest day - sounds like b) is your best and only option.
i prefer to lift after eating anyway. i've normally been fasting all day so i'm typically able to put in a better shift after dinner than before - sort of a TKD thing because i low-carb. the trade off to lifting late isthat you're lifting closer to bed time so if you get particularly pumped up (adrenalin, pre-workout supps) then you're going to have trouble sleeping...