Before you start coming up with arguments about why this isn't a good idea, you should first take some time to get a better understanding of why the owner wants to do this. And I don't mean, "because our competitors are doing it" but what specific advantages the owner expects to get from this.
Often, non-technical people have misconceptions about what advantages and disadvantages there are to technical various strategies and approaches. For example, the owner might assume that moving to this solution will be less expensive overall. Presumably, this solution has some sort of licensing cost, and the owner knows that. So, if they are expecting to save money, they might assume that this will reduce staffing needs. Is that what they are thinking? Or, perhaps the custom site has suffered outages and the owner believes that this solution will make the system more robust. Don't assume you know what they are thinking.
You should find out what ideas about this are bouncing around in the owner's brain. Those ideas may not be well-formed but it's likely more than just that they want to imitate their competitors. Once you have a better idea of these expectations, you can start evaluating whether this solution will address them. Sometimes an off-the-shelf solution increases staffing needs. Maybe you don't need as many developers, but you'll need 10 times more people doing data entry. Does that math work? Maybe not. A lot of companies have adoptingadopted e.g., SAP and found that their total cost of ownership increases dramatically. Some companies have nearly gone bankrupt trying to implement COTS software. Has the owner considered the risk of having the solution provider increase prices in a few years if they are bought by another company looking to extract as much value as possible from the existing customer base? That kind of thing happens regularly. Find out what they are thinking and what they are not thinking about.
Once you understand what they are thinking and start making sense of whether they make sense, you can start asking about things that the owner hasn't considered. Have they considered that having a custom website may allow them to differentiate themselves from their competitors? In other words, how does the owner expect to compete against larger competitors if they constrain themselves to what the larger competitor can do.? It's foolhardy for a destroyer to engage a battleship using battleship tactics. Has your owner considered that their website will no longer be a way to differentiate themselves.?
All of that said, you should, as a professional, take this seriously and consider whether this move makes sense. Try to not 'put your finger on the scale.' Present an honest assessment of what it would mean to move.