How can I speed SQLPS connection when I don't have WMI permissions at the server
Apparentely, after some testing...there is a way to speed up SQLPS load time (tremendously) and prevent the WMI warnings (if you don't need the SQLAS module loaded.

How I tested this on my laptop, was simply by shutting down the WMI service: Stop-Service winmgmt -Force. I then proceeded to edit the module files for SQLPS for my SQL Server 2012 local instance. 

As soon as I load the module initially, I get the same warning messages you show in your question. The one that causes the specific problem is the SqlPsPostScript.PS1 file. This file is a nested module based on the module manifest, so everytime you load SQLPS this script is going to be executed.

This SqlPsPostScript.PS1 file is also why when the module is loaded it changes your location in your prompt. The first line of the file is Set-Location SQLSERVER:. [If you comment out that line it can also speed up the load time for the module a good bit.] 

The lines that are causing the WMI warnings are from the two lines of code that will load the SQLAS module. Which if you look at the documentation for AS PowerShell in BOL it states it is loaded with SQLPS here. It also goes to mention when configuring remote administration for AS that in step 3 WMI Service has to be running. Which if you don't have access to WMI equates to the service not appear as running.

So in the end, as long as you don't need Analysis Services PowerShell module you can edit the SqlPsPostScript.PS1 file, found under:

C:\Program Files (x86)\Microsoft SQL Server\&lt;version&gt;\Tools\PowerShell\Modules\SQLPS\


And comment out the following lines:

$m = Get-Module -ListAvailable | where {$_.Name -eq "SQLASCmdlets"}
if($m -ne $null) { Import-Module $m -Global }


Once I did that and then saved the file, I no longer received the warnings when loading SQLPS.