Calling exe with Start-Process is not getting arguments correctly

in many cases, you can pass a set of parameters/options to an .exe using an array of those options. for instance, the below code works with RoboCopy.exe ... and it is NOT using a splat, it just looks like one.

using Start-Process is not needed at all.

$SourceComputerName = $env:COMPUTERNAME
$Source = "\\$SourceComputerName\c$\Temp\zzz - copy"
    
$DestComputerName = $env:COMPUTERNAME
$Destination = "\\$DestComputerName\d$\Temp\aaa"
    
$FileSpec = '*.*'
    
$TimeStamp = Get-Date -Format 'yyyy-MM-dd_hh-mm-ss'
$Subject = 'RobocopyTest'
$LogFileName = -join ($Subject, '_-_', $TimeStamp, '.log')
$FullLogFileName = Join-Path -Path $env:TEMP -ChildPath $LogFileName
 
$RC_Params = @(
    $Source
    $Destination
    $FileSpec 
    # put your current options below
    "/Log:$FullLogFileName"
    '/NP'
    '/E'
    '/TEE'
    )
 
$RC_Params
#robocopy $RC_Params

Fundamentally, your way of constructing the arguments list is awkward and needlessly complicated (including unnecessary escaping of extraneous ' chars. as `').^[1]

However, the only immediate problem is the mistaken enclosure of the content of string $allArgs in embedded '...', which breaks the call, given that ResourceHacker.exe interprets ' instances literally (it only recognizes " as having syntactic function) and, in general, expects arguments to be passed individually - these behaviors are shared by most CLIs):
To fix this problem, remove `' from $arg1 and $arg8.

Note:

• Likely subsequent problems would be the asynchronous execution of the target executable - due to absence of Start-Process' -Wait switch - in a new window - due to absence of -NoNewWindow - and the inability to query the exit code due to absence of -PassThru and inspection of the returned object's .ExitCode property; additionally, the only way to capture the executable's output would be via files
  (-RedirectStandardOutput and -RedirectStandardError).

• However, all these problems can be solved more simply via direct execution of the target executable, as discussed in the next section.

Taking a step back:

• To synchronously execute console applications (or batch files) in the current console (terminal), call them directly (& $rhPath ... in your case).
  This simplifies the syntax, allows you to directly capture output, as well as to later query their exit code via the automatic $LASTEXITCODE variable.
  Do not use Start-Process (or the System.Diagnostics.Process API it is based on), except in unusual scenarios such as needing to run with a different user identity.
  See this answer and this guidance.

Therefore (note that '...' enclosure of the argument list, as shown in the last command in your question, would be equally mistaken here; for an explanation of the syntactic need to use &, the call operator, in this case, see this answer):

& $rhPath -open $ReplaceTTKVersionInfoRC -save $TTKVersionInfoRes -action compile -log con

You can alternatively use (potentially implicit) splatting if you want to specify the command-line arguments by way of an array (whose definition can easily be spread across multiple lines for readability), along the lines of $rhArgs = @('-open', $ReplaceTTKVersionInfoRC, ...), followed by invocation & $rhPath @rhArgs

Note that, either way, using embedded "..." quoting around individual arguments (as necessary in your $allArgs argument passed to Start-Process) is not required in these cases.

^{[1] Inside "..." (expandable string literals), ' chars. do not require escaping (only " does).
If you want the readability and ease of definition of a multiline string you can use a here-string - which doesn't even require you to embedded " chars. - and subsequently replace its line breaks with spaces, as the following simplified example shows:}

$allArgs = @"  
-username
"$env:USERNAME"
"@ -replace '\r?\n', ' ' # -> e.g., '-username "jdoe"'