You can indeed run LiteFS by yourself, without Consul, as a sidecar / wrapper around your application. We do it in our project and have a Docker Compose example at [0]. In this case, you specify a specific known leader node. We haven't tried getting it running independently with Consul to do leader election / failover.
[0] https://github.com/splitgraph/seafowl/blob/main/examples/lit...