Sounds like it could be an interesting project!
The first thing that really rang my bell here--discounting is generally a no-no. Don't do that, it's a sign of a naive freelancer, especially to people who have been in business. Also anybody who's a savvy negotiator will think--ah ha, we have a total noob here.
There's another possible beginner move I see in the writeup, where it's a bit like you are trying to read their mind, develop a deal that looks good for them, to you.
There's a really common mistake on the introvert side of things (everybody's a mix of introvert/extrovert): Not getting into their heads, and instead staying in your own, and kind of prophesying their needs.
Better to go to them as well, be extroverted in that way--go right to the object, ask for & list _exactly_ what they want, then organize it for them like you'd organize a table of contents. "Is this right? Is that what you're wanting?" Get a definite yes.
You are also planning for them in addition to building for them. This is really important to recognize--that's extra work. I've done it a lot before and A) you need to point that out and B) they may not even want it. Be sure that they do. Document it under project planning or project management.
Pay attention to anything they hesitate about or say they don't need. "It seems like this isn't wanted so much," then emphasize what they do want. Watch for hesitation, call it out, say you want to pay attention to that.
Another introversion-side trap here is: Am I making a mistake? Is my service good enough? Me me me. Look--if you're asking that, probably it's a yes. So watch out for people who clearly don't ask that about their own, say...website. :-) Be careful, that's not about you and your gifts. A huge part of this project is about how good _they_ are with: You, changes in tech, customer responsibilities, etc.
It's smart to be cautious in general with '90s and older-looking sites.
To someone familiar with tech, and especially to a new freelancer, they reflect a big juicy opportunity to show off how far tech has come! A victory for tech and for me!
But to many owners of those sites, there's a secret here behind decades of inattention to tech, and they themselves may not even grasp it:
The situation amounts to another opportunity to delay and stonewall and continually PROVE that tech is a fail. Everybody thinks it's so great, but we don't believe it is! --and thus they can show that modern tech isn't all that great or important.
Pay attention if 1) the business is something more traditional itself 2) the owner uses an older PC or Mac, maybe even a dumb phone 3) they live off the grid, etc.
In these cases--be extra smart by not making this about you. Fit their needs, get out early as a contingency plan, reevaluate and go from there. (This is also a great way to renegotiate--finish initial moves quickly. Then get back in touch for additional work and upgrades projects quarterly, etc. let them know which parts of your services, offerings, prices, terms have changed at those times.) Be really light touch. Tell them tech isn't everything, you know that, and tell them you're trusting them to be up front with you if they just hate tech. This is really crucial to discuss early.
(I've done a _lot_ of these projects myself and this easily applies to 90%; the rest were mostly projects inherited by panicked kids who wanted mom or dad's business website updated ASAP)
Limit both the time and activity scope of your work so you can renegotiate if you discover you're underpaid or things aren't what you thought. Express in writing that you are budgeting hours based on doing this using your best-practice techniques, and that if they request changes, you will be getting in touch to renegotiate.
And again IMO--no discount, show them they are getting your best work and tell them it's coming at your best possible rate.
Hourly billing may also be a mistake, depending--be sure to look at project billing, retainers, and so on. But for now if it works, it's probably fine.
I did my first big solo web project in 2002 for $800, the guy had a website from the 1990s that looked obsolete even then--at time of payment, _after_ delivery, he looked at me like he was getting ripped off, the website was beautiful and functional, I had gone the extra mile, and within 6 months I realized I had a terrible match of a client who hated tech and admitted as much, and the website should have cost him at least $2K. Be really careful about clients who make you feel expensive and who also hate tech--it's not about you, it's on them.
Anyway. A bit didactic here :-) But some thoughts for you to kick around. Good luck.