For me it sounds logical that a character carrying e. g. crops may not use his hands for wielding a two handed weapon. How do you play it?
You should try to avoid applying logic to Descent, if you're at all concerned about game balance, anyway.
Carrying items has no limit (IIRC) in 2E. There's a limit on how many you can equip of certain types, but no limit on how many you can carry.
Objective tokens, specifically, are really reliant on quest-specific rules to detail what can and can't be done with them. However, if the quest-specific rules do not prohibit attacking or taking other actions, then the general rules still apply. The general rules say you can equip a two-handed weapon and attack with it, so you're good.
If carrying a bale of hay required the use of either or both hands (thus preventing the hero from equipping weapons), the quest-specific rules would say so. They don't, FYI.
If you want a logical explanation, perhaps he strapped it to his back. It does take a full action to pick up and also to put down, IIRC. It's far better, in my experience, to come up with logical explanations for how the rules actually work than to spend time trying to change the rules based on what's logical. for one thing, you may find that what the OL considers logical is not what the heroes consider logical.