I am receiving an error when attempting to use a formula without saving the record in the 'In Progress' table (B). However, once I save the record in the 'In Progress' table (B), the same formula works fine.

The record ID (or rid) is not assigned until the record is saved for the first time. 

Don't have a lot of time either, but as everyone says - the record ID does not exist until saved. Luckily, during an add record, when using a custom button on that sheet, you can use the %%rid%% and SaveBeforeNavigating commands to capture the newly saved record ID. Basic button code below, you would just need to sub in your information. This would use a formula rich text field.

var text RID = If([Record ID#]>0, ToText([Record ID#]), "%%rid%%");

var text EditRecord = URLRoot() & "db/" & dbid() & "?act=API_EditRecord"
& "&apptoken= xxxxxx" // if you have the need for Application Tokens enabled.
& "&rid=" & ToText($RID)
& "&_fid_99=now"; // sets this field to current date time but you can set any fields you need as typical 

var text DisplayRecordButMissingRID = URLRoot() & "db/" & dbid() & "?a=dr&rid=";

var text URL =
$EditRecord
& "&rdr=" & URLEncode($DisplayRecordButMissingRID) & $RID; // the %%RID%% does not like being URLEncoded.

//below makes the button. Save before navigating and data-replaceRID=true are the important bits

"<a class='SaveBeforeNavigating' data-replaceRid=true style=\"text-decoration:none; background: #1ba802; border-radius: 5px; color: #ffffff; display: inline-block; padding: 2px 2px 2px 2px; width:200px; text-align: center; text-shadow: none; border: 2px solid #1ba802; font-size: 18px \"href='"
& $URL
& "'>Button Words Here</a>"