Our main tummy time was when we'd lie down on our backs and put him on our bellies. That was so much fun for him, b/c he'd get to look at our faces and interact with us, and of course pay attention to the milk-makers.
About the baby carrier, did you put him in and then get OUTSIDE, to move around, walk briskly, and so on? How many times did you try? Keep trying, get moving, and he might realize he likes it after all.
How much time is he spending on his side?
Is he spending any time on his back? His other side?
Is he uncomfortable on the other side? Have you taken him to a nice, gentle, chiropractor, to have his wee little spine checked out? The tiniest misalignment can cause discomfort and even pain, and having that taken care of with a lovely little gentle adjustment can make WORLDS of difference.
How about cranio-sacral therapy? That's lovely as well. My son's head changed shape within hours of having that done when he was 2 weeks old. It was so cool. I knew it was cool before; the first time I had it done I didn't have to wear my glasses for weeks (I was very blind) and my face felt relaxed for the first time in years (no one came up to me to ask why I was so tense, or to tell me to "smile!" like they usually did)...but to watch DS's head change was astounding. It also helped his latch.
Try those things, see if they help!

About the baby carrier, did you put him in and then get OUTSIDE, to move around, walk briskly, and so on? How many times did you try? Keep trying, get moving, and he might realize he likes it after all.

How much time is he spending on his side?
Is he spending any time on his back? His other side?
Is he uncomfortable on the other side? Have you taken him to a nice, gentle, chiropractor, to have his wee little spine checked out? The tiniest misalignment can cause discomfort and even pain, and having that taken care of with a lovely little gentle adjustment can make WORLDS of difference.
How about cranio-sacral therapy? That's lovely as well. My son's head changed shape within hours of having that done when he was 2 weeks old. It was so cool. I knew it was cool before; the first time I had it done I didn't have to wear my glasses for weeks (I was very blind) and my face felt relaxed for the first time in years (no one came up to me to ask why I was so tense, or to tell me to "smile!" like they usually did)...but to watch DS's head change was astounding. It also helped his latch.
Try those things, see if they help!