Using the existing neutral as a PE grounding conductor on a NEMA 10-30 outlet
We can start with this:

As installed, the grounded conductor for your dryer receptacle is white or gray and cannot be re-identified as an equipment grounding conductor according to the National Electrical Code.


  250.119 Identification of Equipment Grounding Conductors. Unless required elsewhere in this Code, equipment grounding conductors shall be permitted to be bare, covered, or insulated. Individually covered or insulated equipment grounding conductors shall have a continuous outer finish that is either green or green with one or more yellow stripes except as permitted in this section. Conductors with insulation or individual covering that is green, green with one or more yellow stripes, or otherwise identified as permitted by this section shall not be used for ungrounded or grounded circuit conductors.


Then there is using something for which it was not intended. The Code does not allow that.


  Article 110.3(B) Installation and Use.  Listed or labeled equipment shall be installed and used in accordance with any instructions included in the listing or labeling.


The receptacle you have was designed and intended for the third prong to be a grounded conductor not an equipment grounding conductor.


  406.4(C) Methods of Grounding.  The equipment grounding conductor contacts of receptacles and cord connectors shall be grounded by connection to the equipment grounding conductor of the circuit supplying the receptacle or cord connector.


So, if your equipment requires grounding, which it apparently does, you cannot connect a cord connector to the grounded conductor of the circuit supplying the dryer receptacle.

There are other reasons not to do this that I don't have the inclination to list here.

Just run the correct wire and install a proper receptacle for your equipment and sleep soundly.

Good luck!