Given the size of your budget I would start with the following equipment...
- Sound Devices MixPre-3 recorder
- Sony 7506 headphones
- TakStar CM63 mic for indoor dialog ( buy on eBay )
- Sennheiser MKE600 for outdoor dialog
- A good boom pole
- Rycote Lyre Mount Pistol Grip
- "Dead Cats" for both microphones
- Canare Star Quad XLR cables
There are lots of places that sell these effects. I would not try to duplicate the Vintage Electric Bikes ad, but make your own ad, with your own vintage style.
One thing that bothered me a tiny bit is that most real vintage footage from the 60's and 70's is not as sharp as HD footage, so I would have lowered the resolution to either 720p or 480p, and include some shots with intentional soft-focus. Real vintage footage is not crisp and often not always in focus.
If you have a good relationship with your client I would not have a problem with people shadowing me, as long as they don't get in the way. I might even ask them to help me with the shoot by carrying equipment, preparing rooms for shooting, running small errands, etc... The main thing is to look and act like a professional, so your client sees that you are running the show.
My favorite day-pack backpack is the LowePro Photo Hatchback 16L AW Backpack that has hidden camera storage that is only accessible if you take the back pack off. Otherwise it looks like a normal day-pack bag, with no visible camera compartment.
The blue version does not look like a camera bag at all.
There is also a larger version of the same bag : https://goo.gl/UwtWVF
Between these two cameras I would go for the Nikon D5300 because it produces a more detailed video image with better dynamic range.
Otherwise I would try and find a used Panasonic G6 or G7 camera because these cameras produce a better video image than either the Canon or the Nikon, and they can use almost any lens ever made with affordable lens adapters.
I edit with Vegas Pro which has had a high quality PNG export feature for the past 10 years. Really easy to export a frame complete with color corrections, masking, and some FX.