I think you should be able to do it by casting null to the appropiate type. Let’s assume RouteId is an integer:
connection.Execute("spLMS_UpdateLMSLCarrier", new { RouteId = (int?)null, CarrierId = carrierID, UserId = userID, TotalRouteRateCarrierId = totalRouteRateCarrierId }, commandType: CommandType.StoredProcedure);
The problem you are encountering when using regular null is likely that the compiler cannot infer the type of RouteId in the anonymous type when just using null without the cast.