Would you need a remote for the mini amp or does your TV have variable level analog audio RCA or 3.5mm output so you can use the TV remote to control the volume?
Mini amps with remote and analog input for the turntable and digital optical input for the TV plus Bluetooth:
Or a basic stereo receiver from Yamaha with multiple RCA analog inputs.
Onkyo TX-8220 adds digital optical input for a TV and a subwoofer output.
Or for under $100 what is a nearby town or zip code I can search for a used receiver? Reply here or chat.
Bluetooth can be added to most anything with an RCA or 3.5mm input by connecting a $25 Bluetooth receiver or a $35 Echo Dot adds WiFi streaming with voice or app control in addition to Bluetooth.
Turntable: AT-LPW30TK $329 $199.
Plus powered speakers with audio cable input for the turntable plus Bluetooth for streaming audio.
Or better used options:
Or passive speakers and a used stereo receiver:
Used, compare to buying new speakers for over twice the price:
Towers, then you don't need to buy stands:
Or stand mount speakers with included stands are a good value:
Or more stand mount speakers:
Plus a stereo receiver to power the passive speakers:
Optionally add Bluetooth $22 or better WiFi Streaming with voice or app control + Bluetooth $22 or Airplay 2 with Siri + WiFi + Bluetooth $85.
Or an integrated amp with Bluetooth:
Plus speaker wire, pure copper oxygen free 16 AWG or optional thicker 14 AWG. Then How To Strip Speaker Wire and How to Install Speaker Wire.
Or a $40 starter setup with small speakers for a small room https://philadelphia.craigslist.org/ele/d/west-chester-small-inexpensive-stereo/7602743790.html optionally add Bluetooth or WiFi or Airplay 2 from above.
That is getting to be a large room.
Consider the larger Diamond 225 especially if not adding a subwoofer.
Or the dual 5" Monolith towers or even the dual 6" Wharfedlae towers.
Or 6.5" ELAC Debtu B6.2 go lower than some other speakers.
Pass on the basic Sony STR-DH190 unless you really can't spend more than $170. The Onkyo is better if $199 is your limit, it adds digital optical input and subwoofer output over the Sony.
You can add Bluetooth to anything without it by connected a small $25 Bluetooth receiver.
Or better WiFi streamers with voice or app control and they also have Bluetooth:
Echo Dot $22
Wiim Mini or Pro both add Apple Airplay 2 and Siri plus Chromecast and Google Assistant on the Pro.