OP- I actually think you CAN bring Apple Juice for your 1 year old. Here's what TSA says...
"TSA is also modifying the rules associated with carrying breast milk through security checkpoints. Mothers flying with, and now without, their child will be permitted to bring breast milk in quantities greater than three ounces as long as it is declared for inspection at the security checkpoint.
Breast milk is in the same category as liquid medications.
When carrying formula, breast milk, or juice through the checkpoint, they will be inspected, however, you or your infant or toddler will not be asked to test or taste breast milk, formula, or juice. Our Security Officers may test liquid exemptions (exempt items more than 3 ounces) for explosives.
When traveling with your infant or toddler, in the absence of suspicious activity or items, greater than 3 ounces of baby formula, breast milk, or juice are permitted through the security checkpoint in reasonable quantities for the duration of your itinerary, if you perform the following:
1.Separate these items from the liquids, gels, and aerosols in your quart-size and zip-top bag.
2.Declare you have the items to one of our Security Officers at the security checkpoint.
3.Present these items for additional inspection once reaching the X-ray. These items are subject to additional screening.
You are encouraged to travel with only as much formula, breast milk, or juice in your carry-on needed to reach your destination.
You are allowed to bring gel or liquid-filled teethers, canned, jarred, or processed baby food in your carry-on baggage and aboard your plane.
For information on creams, medicines, or other essential items for your child, please read our guidance on these items. Click here for the list of permitted and prohibited items.
Liquids and gels, including baby formula, breast milk, or juice, may be packed in your luggage and checked with your airline.
After clearing security, travelers can now bring beverages and other items purchased in the secure boarding area on-board aircraft.
For more details on navigating the screening process with your children, please read our information on traveling with children."
...so that makes me think that as along as you declare what it is, you should be ok???
As far as food- I have never had a problem. I bring dry food, no liquids (like salsa or jelly) unless they can fit into my plastic baggie and have never ever ever ever been questioned. I've even had protein powder that looks like

and green drink powder that looks like

- no problem at all!! Bring it on the plane!! (or eat it while you're waiting for the plane if you have PB and are worried that the airline will make you throw it out)