Use Powershell to upload to Bitbucket Downloads section - powershell

Question

So we switched to Bitbucket recently - Love the different pricing model / the branch and repo permissions. But, I REALLY miss the Releases feature in GitHub.
BB does however have a "Downloads" section that we can upload recent binaries to. But... It isnt exposed via their API.
I found a nifty script that uses cURL, but I wanted to do this in PowerShell so I dont need any special programs (we use AppVeyor as our CI Server, no cURL).
I have this mostly working - The process is:
Load Signin page to get CSRF token - working
Sign in, redirecting to downloads page - working
Upload file - failing
My code is here
I have isolated the problem to the form data. Here is what it looks like in fiddler when I just do the upload myself:
But when I use Powershell I cannot get it to populate the "WebForms" tab of the request. I've tried using the -InFile and specifying the file to upload, but Bitbucket requires the other shown parameters as well.
How do I specify a series of "Form Fields" to include in the request with Powershell?
EDIT
Adding code here, even though it makes this messy.
$baseUri = "https://bitbucket.org"
$teamName = "xxxxxxxxx"
$projectName = "xxxxxxxxxxxxxxx"
$page = "/$teamName/$projectName/downloads"
$uri = "https://bitbucket.org/account/signin/"
$username = "xxxxxxxxxxxxxxxx"
$password = "xxxxxxxxxxxxxxxx"
$headers = #{
"referer" = "$uri"
}
$json = #"
{
"username":"$username",
"password":"$password",
"submit":"",
"next":"$page",
"csrfmiddlewaretoken":""
}
"#
$formData = #"
{
"token":"",
"csrfmiddlewaretoken":""
}
"#
$body = $json | ConvertFrom-Json
$formData2 = $formData | ConvertFrom-Json
#Load the page once to get the csrf token
Invoke-RestMethod -Uri $uri -SessionVariable mySession -Method Get -OutFile test2.html
#Parse the token
$body.csrfmiddlewaretoken = $mySession.Cookies.GetCookies("https://bitbucket.org")[0].Value
$formData2.csrfmiddlewaretoken = $mySession.Cookies.GetCookies("https://bitbucket.org")[0].Value
#Load the token into the object
$json = $body | ConvertTo-Json
$formData = $formData2 | ConvertTo-Json
#build the uri data needed by bb
$uriData = "username=" + $body.username + "&password=" + $body.password + "&submit=&next=" + $body.next + "&csrfmiddlewaretoken=" + $body.csrfmiddlewaretoken
#Sign into BB now
Invoke-RestMethod -Uri "https://bitbucket.org/account/signin/" -WebSession $mySession -Body $uriData -Method Post -OutFile test.html -Headers $headers
#Update referer to the downloads page
$headers.referer = $baseuri + $page
$fulluri = $baseuri + $page
#Upload the file
Invoke-RestMethod -Uri "$fulluri" -Method Post -OutFile test3.html -WebSession $mySession -Headers $headers -Body $formData -ContentType "multipart/form-data"