Short answer
Give elected officials a short, regular update on milestones, budget, risks and decisions, in plain language and with no surprises. Confidence comes from hearing about problems early from staff, together with the plan to fix them, rather than from a vendor or the news.
Council members do not need to understand the chart of accounts design, but they do need to know whether the project is on time and on budget, what decisions are coming, and what risks you are watching, and in my experience a one-page update every quarter does more for confidence than a long presentation once a year. I would use the same format every time so they can see trends, and I would tie the project back to the outcomes they approved it for, like faster reporting, better budget visibility or easier service for residents.
The most important habit is to share bad news early, because elected officials can handle a delay or a change order when they hear it from staff along with a plan, but they lose trust quickly when they learn about it from a vendor, an auditor or a reporter. An independent assessment at major milestones can also help, since it gives the board a view that does not come from the people running the project.
Finally, celebrate the wins they can see, like the first clean month-end close or the first budget built in the new system, since a long project needs visible progress to keep its support. The Start at Zero Substack covers how other agencies keep their boards with them, and subscribing is free.