My volume is not working well on the MX-350 remote, and I need to be very close to the TV, despite having the RF base station fully working. Thanks!

Yes, buy a remote control :>) They are readily available, no need to program and can be obtained from Amazon, eBay and specialty remote retailers like www.remotes.com

Google

Seiki (model) replacement remote

https://www.google.com/search?q=seiki+tv+replacement+remote

The most likely reason for your issue is a dirty button and the contacts. Switch Contact Cleaner may fix it.
..

Make sure that all the batteries are good!!
This procedure outlines the typical process to pair the sensor and base.
Base Station Battery Installation
·Remove the printed vinyl label from the LCD screen of the base station
·Position the remote sensor near the base station
·Open the battery compartment at the back of the base station and insert (2) AA batteries according to the polarity markings. Replace the battery compartment cover.
·Within a short time the RF signal icon (located to the left of the outdoor temperature channel indicator) will start flashing. This indicates that the base station is ready to pick up the RF outdoor temperature signal from the remote sensor. It will continue to flash for 3 minutes or until the signal from the remote sensor is detected.

Register the Remote Sensor
·Open the battery compartment at the back of the remote sensor and slide the channel selector switch to your desired channel. For the first remote you may select any channel, for additional remotes select any unused channel
·Insert (2) AAA batteries according to the polarity markings. Be sure to install the batteries in the remote within the time that the RF signal icon is flashing on the base station. 1541 WeatherGuide Wireless Weather Forecaster with Barometer

Base Station Battery Installation
·Remove the printed vinyl label from the LCD screen of the base station
·Position the remote sensor near the base station
·Open the battery compartment at the back of the base station and insert (2) AA batteries according to the polarity markings. Replace the battery compartment cover.
·Within a short time the RF signal icon (located to the left of the outdoor temperature channel indicator) will start flashing. This indicates that the base station is ready to pick up the RF outdoor temperature signal from the remote sensor. It will continue to flash for 3 minutes or until the signal from the remote sensor is detected.

Register the Remote Sensor
·Open the battery compartment at the back of the remote sensor and slide the channel selector switch to your desired channel. For the first remote you may select any channel, for additional remotes select any unused channel
·Insert (2) AAA batteries according to the polarity markings. Be sure to install the batteries in the remote within the time that the RF signal icon is flashing on the base station.
1541 WeatherGuide Wireless Weather Forecaster with Barometer

Check this link to download the user manual to your remote control. It has all details.

I believe DirectTV has excellent customer service reps that would be delighted to help you!

Depending on the age of the VCR it will have either a small screw on the back near the aerial socket, or the adjustment will be on a onscreen menu somewhere. It will be called the "RF OUTPUT".
They way to do them is for the screw type - turn it slightly - then with a video playing (something that you can tell what to look for) retune your TV. You might need to do this several times to get clear of stations.
With the menu type you just need to enter a RF number. This is often the best type as most VCR's give you the full RF range. Whereas the screw will only be a short range. Enter a RF frequency number which will be well away from broadcast stations. You can find which TV stations broadcast on the frequences in your area by looking it up in Google. Once entered, play a video and retune the TV to it.
Some VCR's allow the output to be switched off. You can use this option if you have the VCR connected via a scart lead. As you don't need to have the TV tuned in to the VCR.

Replace remote battery with a new one. It may not be transmitting enough power to work more than 6 feet from the TV.

go online and google it

Toshiba TV 2151TB Tuning
Press remote button adjacent to the picture brightness button.
The symbol of a clock appears.
Press the button again and the >>> symbol appears, this is for course tuning.
Touch the Increase Volume button and auto tuning starts.
When a station is found two green underlines appear.
Type the station number i.e.for 2 type 02, this stores the station.
Touch the Increase volume button again to find the nexr station.
Upto 39 stations can be stored.
Fine tuning of any station is achieved by pressing the tuning button until three blocks sysbol is displayed.
Use the Increase/Decrease Volume buttos to fine tune the station.

With the multiple TV's what you are looking to do is called "IR routing" There is not exactly a "pre-packaged" solution.

Try one of these companies I think all of them can handle this:
http://www.crestron.com/
http://www.universalremote.com/
http://www.amx.com/
http://www.rticorp.com/

I would have it professionally installed. You can find a professional in your area using the locater on the websites.

Hope that helps!