Вот лог после обновления. Похоже, изменений особых нет...

C:\Program Files (x86)\Crypto Pro\CSP>csptest -tlsc -server api.stage.mdlp.crpt.
ru -v -v -proto 4
12 algorithms supported:
Aglid Class OID
[00] 0x660e 0x6000
[01] 0x6610 0x6000
[02] 0x6801 0x6000 1.2.840.113549.3.4 (rc4)
[03] 0x6603 0x6000 1.2.840.113549.3.7 (3des)
[04] 0x6601 0x6000 1.3.14.3.2.7 (des)
[05] 0x8003 0x8000 1.2.840.113549.2.5 (md5)
[06] 0x8004 0x8000 1.3.14.3.2.26 (sha1)
[07] 0x2400 0x2000 1.2.840.113549.1.1.1 (RSA)
[08] 0xaa02 0xa000 1.2.840.113549.1.9.16.3.5 (ESDH)
[09] 0xae06 0xa000
[10] 0x2200 0x2000 1.2.840.10040.4.1 (DSA)
[11] 0x2203 0x2000
Cipher strengths: 256..256
Supported protocols: 0x80:
Transport Layer Security 1.0 client side
dwProtocolMask: 0x800a0aaa
Protocol version: 3.1
ClientHello: RecordLayer: TLS, Len: 125
SessionId: (empty)
Cipher Suites: (c0 14) (c0 13) (00 35) (00 2f) (c0 0a) (c0 09) (00 38) (00 32) (
00 0a) (00 13) (00 05) (00 04)
130 bytes of handshake data sent
0000 16 03 01 00 7d 01 00 00 79 03 01 5e 03 62 6f 40 ....}...y..^.bo@
0010 b7 ca d6 e8 44 d2 b8 6e 3e 74 ca 3c da 27 1b 82 ....D..n>t.<.'..
0020 e0 a1 04 58 d0 37 13 66 cc eb d3 00 00 18 c0 14 ...X.7.f........
0030 c0 13 00 35 00 2f c0 0a c0 09 00 38 00 32 00 0a ...5./.....8.2..
0040 00 13 00 05 00 04 01 00 00 38 00 00 00 1b 00 19 .........8......
0050 00 00 16 61 70 69 2e 73 74 61 67 65 2e 6d 64 6c ...api.stage.mdl
0060 70 2e 63 72 70 74 2e 72 75 00 0a 00 06 00 04 00 p.crpt.ru.......
0070 17 00 18 00 0b 00 02 01 00 00 17 00 00 ff 01 00 ................
0080 01 00 ..

7 bytes of handshake data received
0000 15 03 01 00 02 02 28 ......(

**** Error 0x80090326 returned by InitializeSecurityContext (2)
An error occurred in running the program.
WebClient.c:633:Error performing handshake.
Error number 0x80090326 (-2146893018).
Получено непредвиденное сообщение или оно имеет неправильный формат.
Total: SYS: 0,031 sec USR: 0,000 sec UTC: 0,094 sec
[ErrorCode: 0x80090326]

C:\Program Files (x86)\Crypto Pro\CSP>

В реестре по указанному Вами пути:

Functions REG_SZ
TLS_ECDHE_RSA_WITH_AES_256_CBC_SHA384_P256,
TLS_ECDHE_RSA_WITH_AES_256_CBC_SHA384_P384,
TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA256_P256,
TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA256_P384,
TLS_ECDHE_RSA_WITH_AES_256_CBC_SHA_P256,
TLS_ECDHE_RSA_WITH_AES_256_CBC_SHA_P384,
TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA_P256,
TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA_P384,
TLS_DHE_RSA_WITH_AES_256_GCM_SHA384,
TLS_DHE_RSA_WITH_AES_128_GCM_SHA256,
TLS_RSA_WITH_AES_256_GCM_SHA384,
TLS_RSA_WITH_AES_128_GCM_SHA256,
TLS_RSA_WITH_AES_256_CBC_SHA256,
TLS_RSA_WITH_AES_128_CBC_SHA256,
TLS_RSA_WITH_AES_256_CBC_SHA,
TLS_RSA_WITH_AES_128_CBC_SHA,
TLS_ECDHE_ECDSA_WITH_AES_256_GCM_SHA384_P384,
TLS_ECDHE_ECDSA_WITH_AES_128_GCM_SHA256_P256,
TLS_ECDHE_ECDSA_WITH_AES_128_GCM_SHA256_P384,
TLS_ECDHE_ECDSA_WITH_AES_256_CBC_SHA384_P384,
TLS_ECDHE_ECDSA_WITH_AES_128_CBC_SHA256_P256,
TLS_ECDHE_ECDSA_WITH_AES_128_CBC_SHA256_P384,
TLS_ECDHE_ECDSA_WITH_AES_256_CBC_SHA_P256,
TLS_ECDHE_ECDSA_WITH_AES_256_CBC_SHA_P384,
TLS_ECDHE_ECDSA_WITH_AES_128_CBC_SHA_P256,
TLS_ECDHE_ECDSA_WITH_AES_128_CBC_SHA_P384,
TLS_DHE_DSS_WITH_AES_256_CBC_SHA256,
TLS_DHE_DSS_WITH_AES_128_CBC_SHA256,
TLS_DHE_DSS_WITH_AES_256_CBC_SHA,
TLS_DHE_DSS_WITH_AES_128_CBC_SHA,
TLS_RSA_WITH_3DES_EDE_CBC_SHA,
TLS_DHE_DSS_WITH_3DES_EDE_CBC_SHA,
TLS_RSA_WITH_RC4_128_SHA,
TLS_RSA_WITH_RC4_128_MD5,
TLS_RSA_WITH_NULL_SHA256,
TLS_RSA_WITH_NULL_SHA,
SSL_CK_RC4_128_WITH_MD5,
SSL_CK_DES_192_EDE3_CBC_WITH_MD5

Спасибо большое, Максим!
Завтра обязательно попробую!

Что значит не верен?
Там Криптопро CSP. Он по умолчанию игнорирует список в Client hello и всегда использует ГОСТ.