All experiences vary, so take this with a grain of salt. I'm writing this from my personal experience of 2 kids.
It will be fine, especially if this is your first. The first 6 months to a year are the hardest. Once the baby is sleeping through the night, and especially once you start daycare, you will have a normalish schedule. You must conform to a very strict 9-5 style schedule for both work and professional development. This will force you to be productive during those hours and then hard-stop. Evenings will be a crapshoot. Sometimes you'll have the energy, mostly not.
Just recently indeed.com scraped our website and posted a job on our behalf. My email address started receiving resumes for our IT services division and I figured they were in the loop so I trashed them.
To clarify, any clients I have that behave that way are firmly out of the SMB segment (or have grown to that point as we have worked together), seem to be in cash-flow heavy industries like manufacturing or construction, and we've been working together for several years.
I think you made the point I was going for - it may be more hassle than it's worth to do work for a client that makes a fuss over $2200 vs $2000.
I think web-design in the SMB market is completely commoditized between website builders and even CSS toolkits like Bootstrap lowering the barrier to entry. I barely know CSS and will just buy a $20 design from wrapbootstrap and the client doesn't care.
Anything related to web-design should be a side-effect of the true product you're selling, which in your case should be web-based software application development (IMO).
My opinion:
#1) Get really good at a web-framework. Doesn't matter which, but something like laravel would be a great choice. Learn MySql/MSSQL to go with it.
#2) Market yourself as a specialized consultant and bill $100 hr+. I don't know what market you're in, but if this is a big leap for you then work up to it. The kind of clients you want to work with will pay this without blinking. Sometimes they won't even know about the bill because it goes straight to AP. blah blah blah patio11.
#3) Never say you do web-design. Say, 'I convert your MS Access application that runs a critical component of your business (barely) into a multi-user/role, always-online, accessible-from-home web-application. Along the way we're also going to streamline this to make your company more money. It will pay for itself. That will be $30k.'