Let's say they are 7ft 4in tall, and cast their innate Disguise Self, using the rules from Mordenkainen's Monsters of the Multiverse:
Firbolg Magic • MotM, pg. 15
Once per long rest, you can cast detect magic and disguise self. When you use this version of disguise self, you can seem up to 3 ft. shorter or taller. You can also cast these spells using any spell slots you have. Your spellcasting modifier is either Intelligence, Wisdom, or Charisma (choose when you select this race).
That means they may appear to be 10ft 4in, and then after the spell is cast they decide to lift a barrel above their head (e.g. the barrel is not a 'belonging on your person' when the spell is cast). That means the hands are ~15ft foot above them (~10ft + half height for arms), but the barrel is more like ~10ft above them (~7ft + half height for the arms).
Normally I couldn't see this being an issue for a shorter character, especially as normally the Disguise Self spell only makes you taller/shorter by 1ft:
You make yourself--including your clothing, armor, weapons, and other belongings on your person--look different until the spell ends or until you use your action to dismiss it. You can seem 1 foot shorter or taller and can appear thin, fat, or in between. [...]
So for a shorter character the discrepancy wouldn't be that great. At worst, maybe 10½ft, (if a 6ft character gained 1ft, and lifted the barrel up an extra 3½ft up), vs 9ft (6ft+3ft), which is probably smaller than the object being lifted... so I can imagine being the illusion covers up this up. But even if the barrel is ~3ft in diameter, there is too much to 'cover up' without the spell doing more than it says it does and either the barrel ends up floating between their arms or invisible if it is supposed to appear 2-5ft higher up than it is.
How should I align the rules and my intuition/limited suspension of disbelief on this set up? Or is this a case by case GM problem?