OP, it sounds very much like he's wanting you, not so much the milk necessarily, and as you say he's still settling at nursery so being on your breast is the place he's feeling happy, safe and loved. It's really, really common when babies start to have to separate from you like this, for nursery or childcare, for them to be very clingy when they're back with you. It WILL settle down but unfortunately it could take a few months - or it may be really quick, it depends on your own baby's personality.
You sound like you're doing exactly the right thing by letting him feed, letting him be with the breast. He's getting what he needs and if you can cope with the extra franticness (which can be worrying and wearing) it will settle as he realises that you WILL come back EVERY TIME!
It doesn't sound to me like the problem is your milk supply, and as you're feeding at lunchtime it's unlikely to be affected by being away for the day. He's not a newborn, and your milk supply will be very resilient now. Your breasts will continue to make milk even when they feel empty and just letting him switch sides if he wants to will maximise his milk access.
Teething is another one in the mix and this will also be really helped with BFing, so yes, this may be a significant factor. The act of suckling will reduce his pain, whether there's milk pouring out or not. Have you considered a teething necklace? They work really well for some babies and can be used with all the others suggestions above.
I understand that you are worried about supply because he doesn't eat too many solids. He's likely to eat a lot more when you're not there, and possibly eat virtually no solids when he's with you. This will settle down and your breasts WILL cope.
You sound like you're doing things perfectly. How are you coping with his demands? It can be exhausting.