You can file IFR? if the plane is not IFR certified? do you put "keep me clear of clouds in the remarks"?
FAAO 8700.1 Ch. 8
9. USE OF AIRCRAFT NOT APPROVED FOR IFR OPERATIONS UNDER ITS TYPE CERTIFICATE FOR INSTRUMENT TRAINING AND/OR AIRMAN CERTIFICATION TESTING. The following paragraphs are intended to clarify the use of an aircraft not approved for IFR operations under its type certificate for instrument flight training and/or airman certification testing.
A. IFR Training in Visual Meteorological Conditions (VMC). Instrument flight training may be conducted during VMC in any aircraft that meets the equipment requirements of part 91, 91.109 , 91.205 , and, for an airplane operated in controlled airspace under the IFR system, 91.411 and 91.413 . An aircraft may be operated on an IFR flight plan under IFR in VMC, provided the PIC is properly certificated to operate the aircraft under IFR. However, if the aircraft is not approved for IFR operations under its type certificate, or if the appropriate instruments and equipment are not installed or are not operative, operations in Instrument Meteorological Conditions (IMC) are prohibited. The PIC of such an aircraft must cancel the IFR flight plan in use and avoid flight into IMC.
B. Type Certificate Data. Appropriate type certificate data will indicate whether the aircraft meets the requirements for IFR operations.
(1) Section 91.9(a)prohibits aircraft operations without compliance with the operating limitations for that aircraft prescribed by the certificating authority.
(2) Section 91.9(b)prohibits operation of a U.S.-registered aircraft requiring an airplane or rotorcraft flight manual unless it has on board a current and approved airplane or rotorcraft flight manual or approved manual material, markings, and placards containing each operating limitation prescribed for that aircraft.
So yeah, no IMC is allowed with the DA-20.
That said, based upon the OP's logic in buying either a DA20-A1 or older C172, I don't think that an LSA which is IFR approved/equipped is anywhere near the same price range. The Tecnam P2002 Sierra, the low wing model, which I've seen flown IFR, is probably going to push 140-150K by the time you get it set up for IFR, so that is probably going to cost way to much. I don't think you can find an S-LSA out there, even used, at this point for under 70,000.