Can I use Microsoft Power BI data visualisations and dashboards with Solver budgeting and forecasting data?

Yes, Solver has a pre-built connector for Power BI.

Solver integrates with Power BI dashboards to provide reporting, budgeting, and data warehouse functionality that, combined with Power BI, provides a world-class enterprise solution for planning, reporting, and analysis.

Best of Both Worlds: the image below shows the features and functionality of Power BI and Solver.