Can enquiring (merisik), proposing and the engagement be done together at one event to save costs?

Yes. The Shariah only requires a clear proposal and consent; holding merisik, the proposal and the engagement at separate events is a custom, not a religious obligation.
In Islam, khitbah is a request to marry accepted by the woman and her guardian; the Prophet ﷺ himself proposed simply, through a representative. Merisik aims to make sure the woman is not already promised to someone and is suitable, and this can be confirmed earlier through family conversations or a phone call. Many Malaysian families now combine all three in one visit: enquiring, stating the intention, agreeing on the length of the engagement, the hantaran and the date, and putting on the ring by the mother or a female relative. Make sure both families agree to the simpler approach so no one feels slighted. The money saved is better spent on starting the household. See the etiquette of merisik and Proposal (khitbah).