inn more recent versions the concept of embeddables and embeddable containers got introduced. An embeddable is everything that can be shown within the UI of something else (e.g. a visualization, a control, ...), and an embeddable container is something which can show one or multiple embeddables. The dashboard itself is such an embeddable container as it can show multiple embeddables (mostly visualizations). But a dashboard is also an embeddable itself, which means if there is a plugin that implements the dashboard container interface, it can render a dashboard which sounds like exactly the thing you need in your plugin.
In the Kibana repo there is an example plugin in
test/plugin_functional/plugins/kbn_tp_embeddable_explorer that embeds a dashboard using the embeddable API. The most interesting parts are in