Optech is highly rated and that looks like a great fit on the D3300.
There just isn't a cheap way to get close - I've been trying... LOL.
Throwing money at those $300 800mm lenses is literally flushing money down the toilet.
To be clear, your D3300 has a "crop multiplier" ( don't fight me on the term photo-snobs, you know what I mean) of 1.5x
So, your 200mm is really 300mm in "normal" terms.
Options that you could consider
a) Tamron 70-300. That would get you 450mm. Can be found for $350 refurb
b) Nikon 70-300. The Tamron is said to be sharper at the long end, the Nikon is sharper everywhere else.
c) Find a used 80-400.
d) Save money and buy the Tamron or Sigma 150-600
e) Save money and buy the Nikon 200-500
f) You have more money than I do, so get the 600 f/4 and be done with it.
Unless you have a fast lens to start with ( f/2.8 constant or f/4 constant), don't consider teleconverters. You'll want to, and I'm telling you don't. Your camera won't focus, you lose light, and your camera won't focus.
No matter what you get, it won't be enough. You just never have enough reach for birds. Ever never. I've gone out with birding clubs where the typical setup is some huge Canon body with a literally comic sized 600 f/4 or 400 f/2.8 with 1.7 teleconverters and they complain.
I'm try gimp by with a 70-200 f/4 and a 1.7TC on my daughters crop body D5200 = ~ 510mm.
Patience and persistence will eventually reward you with decent shots.
If you're super patient and persistent, you can get a more than a few decent shots.
When you see those "OMG AWZSOME BIRD PHOTO" photos, those guys are not using $400 bodies and $300 lenses.
They've got $6000 Mark KDIIII or D4S with $7500 600f/4 lenses --- all those pieces focus 10 times faster than what we have and have optics to back it up.
I'm telling you all this because I've recently got into birding and JFC .. I can get "good" shots, but I'm sitting next to people with this gear and they post Holy SH~t shots. Same light, shoulder to shoulder damn near same exposures.
For me, the following is one of my good shots - but it's still soft, lacks contrast and is generally "average".
https://www.flickr.com/photos/coustier/23902666152/in/dateposted-public/