If you are satisfied with your smart tv wi-fi connection and what it provides then there is no need to enable the wi-fi on the blue ray player. When you set up your blue ray player you will just bypass the option to set up the internet access. The blue-ray player can just be used to play your dvd movies. I have two of these players. One is used in the master bedroom with the wi-fi access and the other one is in my media room and only used to play movies. I use a roku wired connection in the media room when I want to stream movies to the projection tv.
One thing to note in your main question on wi-fi competition is that the more devices you have in your home active with wi-fi can cause drops in what you are streaming. If you were streaming movies through your tv in one room and streaming movies through a smart blue ray player in another room to a non smart tv then you could see occasional drops due to the wi-fi contention.
If you want to set up your blu ray player without an internet connection I would suggest turning it on initially via the setup to receive any updates that are available. You would then turn it off via the setup. If you have issues in the future with the dvd player then use setup to turn the internet connection back on to check for updates to the player which will be downloaded and installed for you. After that you can disable your connection via the setup.
If you are using the wifi on both at the same time then the wifi strength for both items would be less. So if your internet is 3mb and they are both using the wifi at the same time then each would use approx. 1.5 mb of that total wifi depending on distance from wireless router and each devices wireless card. If one is off or on but not using the wifi then the other would use all the wifi bandwidth. I have been downloading stuff on my computer and so the wireless connection for the BluRay player was down to about 2 Mb/s and I still had very good quality streaming with no hiccups when using Amazon Prime.
You shouldn't have any problems, you're not going to be streaming with your TV and blu ray player at the same time. Leave the settings to "update" when "available" on your blue ray player don't worry about turning anything "off". These systems do not update that often and you will never realize when they do. I've had my kids on both their laptops steaming netflix and I was streaming movies on my blu ray player with no problems. All depends on how fast your internet service is to your residence.
Both system have built in wifi. It is your choice as to which item you want to use for that service. You can use either the TV or BR Player. They can not 'compete'. The blue ray has to be turned on and they you choose the service from there, the same way you do with the tv.