I see a couple of issues here...
I don't see what your plan is for splicing into the 12VDC and 120VAC side of the Inverter.
You can tap the 12VDC by just paralleling the 4AWG cables. The problem for this is finding a suitable place to tap the 4AWG cable. It appears your 4AWG cables are trimmed and inserted into a LUG arrangement. You cannot double up on one of these LUGS. Some CONVERTERS have DUAL LUG connection points but I am not seeing anything like that inside the available connection points. You need to run 4AWG cables all the way to the NEW INVERTER.
The next area of concern is where to connect the 120VAC OUTPUT wiring. If your plan was to use the same 120VAC Receptacles for the items you want to have on emergency power then you have to install a switch or contactor that will select either the 120VAC power coming from the Distribution panel or coming from the Inverter. You cannot parallel these leads like you did on the 12VDC cables. It has to be selected from one of the two 120VAC sources. This is more complicated to do and is the big reason why alot of folks will just provide a drop cord arrangement coming from the INVERTER 120VAC and in order for you to use it you have to plug your 120VAC appliances into the emergency 120VAC receptacle.
This is what I did with my 600WATT Pure Sine Wave INVERTER. It is installed on the inside cabinet wall as close to the BATTERY BANK Selector switches using the ring terminal lugs to make my 12VDC connections. The INVERTER must also have a pretty heavy fuse block installed as well. This needs to be the same type of battery studs so that all 4AWG cabling uses the ring terminal ends for all 12VDC stud connections. With all of this in place then you can just plug in a HD 120VAC Extension cord into the PSW Inverter 120VAC receptacles and run this cord to the location of the 120VAC appliances you want to feed. Since there is usually two 120VAC receptacles on the Inverter I ran an extension cord to the home entertainment area and a second extension cord to the bedroom night table location. Using the Pure Sine Wave type Inverter I do not have to worry about what I plug into the emergency power receptacle.
This is a drawing I did up showing a typical install using the two Extension cords idea.
If you really want to Install a manual SELECTOR TYPE switch to use your existing 120VAC Receptacles then you could do something like this... Note All 120VAC connections are inside approved type Electrical Junction boxes.
These type of 120VAC selector switches are not cheap...
just some of my thoughts to give some additional ideas. Working with the 120VAC side of things has to be installed in a safe professional way...
This is my floorplan for my off-road POPUP setup showing where the Inverters and extension cord multi-outlets are located.
Roy Ken