There are a lot of variables you are not accounting for.
First answer the following questions:
Are you using two different ISPs or the same one with dual links?
Why are you trying to use two different ISPs? Redundancy, performance, etc?
Are you just directly assigning public IPs to the two interfaces on the docker host?
What other networking gear do you have?
How are you defining your DNS entries?
Off the top of my head, the real professional/enterprise solution for this is as follows:
1 - Purchase your own public IP space (min is /24 I believe)
2 - Get two ISPs that will advertise your space and will peer with you (normally done with BGP)
3 - Purchase a router and/or a firewall (a real router/firewall not a best buy one; example Cisco or Palo Alto) that will large enough to learn all the internet routes from your ISP
4 - Peer with both ISPs and advertise your public IP address space out both ISPs at the same time
5 - Assign your docker host with a single public IP or NAT the internal address to one of your public IPs
6 - You are done!
The result of this will be the following:
1 - Users that are closer on the internet (not physically closer) will route through the closer ISP. Just because your DC is next door to Starbucks, doesn’t mean Starbucks is closer on the internet than the McDonalds 2 miles away. This happens due everyone having different ISPs. Those ISPs have to peer together and sometimes that peering point is in a different state/country.
2 - If one of your ISP links goes down, all traffic will automatically flow in and out the other. When it comes back up, it will automatically go back out both.
3 - You will be load balancing between the two ISPs depending on where your users are sourcing
I work as an network engineer and this is what I do on a daily basis. Granted I may have missed some finer points, no one is perfect. This solution is not cheap and you may be better off looking for a cloud hosting provider. Based on what you are trying to host / do, I would look into DigitalOcean. They take care of these things in the background so you don’t have to worry about it.