There are multiple license styles at play here;
- MIT/BSD/Apache-style license style
MySQL connector, assuming you mean MySQL Connector/J, is dual-licensed; You can either choose the commercial license (you will have to pay for this), or the LGPL license. Licensing is a thing the authors of a product do; you can't change the license. If the task at hand demands a certain license, you ask the authors if they offer it, and how much you need to pay to get this; if it's too much for your wallet or this license is not available, your only resort at that point is to either change the requirements, or, remove that product from your codebase entirely, and e.g. switch to PostgreSQL (a different DB engine), for example.
Note that LGPL isn't the same as GPL. It's more like MIT/BSD/Apache-style licenses than GPL is, so one way out may simply be to ask your client's lawyers if LGPL is acceptable.
For future consideration, always check with the client what kind of license restrictions they have. For example, if MIT/BSD/Apache is the only way, then you cannot use MySQL here. Period. MySQL is not licensed like that and Oracle (the owner of MySQL) doesn't offer it. In that case, it's possible a commercial license would have sufficed (so, make sure to charge the client to take into account you'll need to acquire a commercial license). Alternatively, take into account that you'll need to spend some time learning about using different tools (such as PostgreSQL, whose license is MIT/BSD/Apache-style), and bill accordingly.
If you need more information about whether or not the LGPL of MySQL and MySQL Connector/J is compatible with your client's needs, or what's going to happen if you ignore either a license or your customer's written requirements, you should ask a lawyer.