You have to understand that form a certain perspective there is no such thing as a bad logo. The distinctiveness and the recognition, the values etc are all things that are made by marketing rather than the logo itself.
Believe or not the now iconic Nike logo was not really appreciated by the powers to be. That is because theres nothing superior about the logo. However years of advertising work has made the logo iconic.
This said a logo can have disadvantages:
- It can have a aspect ratio that makes it hard to use in many contexts
- It can rely too much on certain ambient factors such as specific color that can not be printed. This can be problematic if you need to put the logo onto a piece of merchandise etc.
- Logo can be expensive to print
- Logo can be inappropriate
These can be fixed, but re branding would mean the need to rethink your organization. So first make the website work, that can be done with any logo. Then when you've shown you can do that I'm sure they will ask you to re brand their marketing material if they need it. Otherwise it sounds more like saying "well i could not play the cheap synthesizer but I'm sure if you get me a grand piano id be great". Ultimately its up to the client to say if they want more done or not.
Its possible to have multiple brands?
It has certain downsides. The primary downside is that now your marketing budget has to do two times more than before with the same amount of cash. So for a very small organization strapped on cash it is not ideal.
But then if the choices are limited it might be the only way ahead in the world. Rebranding itself may be useful in context if the company is trying to signal that they have made a change in the strategy as it can signal belief in a bright the future.
So should they rebrand?
In a way its natural to think you can do a better job. And I'm sure you can. However its not a skill to make a good graphical design with world class resources at your disposal. The trick is rather making something really spectacular without those limited resources you have.
The question of rebranding really has to come from the client. Theres not much use in just rebranding if your really not refrcusing your marketing efforts or changing the product. The aim of doing these things is to sell more, changing the decor certainly can do this. But its not guaranteed to do this. Its no use to be hip and cool if their product is not for the hip and cool.