I haven't had time until tonight to play with my remote problems again, but the answer did dawn on me and I found the solution, in case anyone else out there is having this problem (and maybe the Pluto staff could integrate what I found into the next release).
The reason my codes where showing up in the log as unrecognized was due to the fact that in a table called RemoteMapping in database pluto_main, there is a list of all the acceptable names for remote codes from IR remote configurations. As it turns out, many of the names given to the buttons in the stock Hauppauge 350 config are not included in RemoteMapping. All I had to do was add the names in the right places (or, changing the Hauppauge 350 config to match the names in the database would work too).
Here is the text of my sql file to fix the RemoteMapping table and now the Hauppauge 350 remote works splendidly (my only complaint is that there does not seem to be a message available to go directly to the Videos screen, but I think I might be able to find that and add it):
-- MySQL dump 9.11
--
-- Host: localhost Database: pluto_main
-- ------------------------------------------------------
-- Server version 4.0.24_Debian-10
USE pluto_main;
DROP TABLE RemoteMapping;
--
-- Table structure for table `RemoteMapping`
--
CREATE TABLE `RemoteMapping` (
`PK_RemoteMapping` int(11) NOT NULL auto_increment,
`Description` varchar(40) default NULL,
`ScreenType` char(1) default NULL,
`RemoteLayout` char(1) default NULL,
`Mapping` text,
`psc_id` int(11) default NULL,
`psc_batch` int(11) default NULL,
`psc_user` int(11) default NULL,
`psc_frozen` tinyint(1) default '0',
`psc_mod` timestamp(14) NOT NULL,
`psc_restrict` int(11) default NULL,
PRIMARY KEY (`PK_RemoteMapping`),
UNIQUE KEY `psc_id` (`psc_id`)
) TYPE=InnoDB;
--
-- Dumping data for table `RemoteMapping`
--
INSERT INTO `RemoteMapping` VALUES (1,'Default',NULL,NULL,'Power;|0 -203 1 194\r\n1;One;|0 -203 1 28 26 16\r\n2;Two;|0 -203 1 28 26 17\r\n3;Three;|0 -203 1 28 26 18\r\n4;Four;|0 -203 1 28 26 19\r\n5;Five;|0 -203 1 28 26 20\r\n6;Six;|0 -203 1 28 26 21\r\n7;Seven;|0 -203 1 28 26 22\r\n8;Eight;|0 -203 1 28 26 23\r\n9;Nine;|0 -203 1 28 26 24\r\n0;Zero;|0 -203 1 28 26 25\r\nenter;|0 -203 1 28 26 5\r\nSkipNext;SkipForward;skipfwd;|0 -106 1 42 5 +30\r\nSkipPrior;Replay/SkipBackward;skipback;|0 -106 1 42 5 -15\r\nchup;Ch+;ChanUp;|0 -106 1 65 5 +1\r\nchdown;Ch-;ChanDown;|0 -106 1 65 5 -1\r\nfastforward;Forward;|0 -106 1 41 43 +2\r\nRewind;|0 -106 1 41 43 -2\r\nPause;|0 -106 1 39\r\nPlay;|0 -106 1 37\r\nMute;|0 -106 1 97\r\nStop;|0 -106 1 95 203 1\r\nDown;|0 -203 1 201\r\nUp;|0 -203 1 200\r\nLeft;|0 -203 1 202\r\nRight;|0 -203 1 203\r\nok;|0 -203 1 190\r\nvoldn;VolDown;|0 -106 1 90\r\nVolUp;|0 -106 1 89\r\nrecorded_tv;RecTV;|0 -203 1 401 29 5\r\nDVD;|0 -106 1 43\r\ntv;live_tv;LiveTV;|0 -106 1 43 29 1\r\nBack;Back/Exit;|0 -203 1 240\r\nClear;|0 -203 1 363\r\nepg;Guide;|0 -203 1 126\r\nstart;Go;Home;|0 -203 1 5 3 <%=M%>\r\nMusic;|0 -203 1 401 29 4\r\nRadio;|0 -106 1 43 29 6\r\nRecord;|0 -106 1 102\r\ndisplay;Menu/i;info;list;|0 -203 1 399\r\ngreen;|0 -203 1 28 26 177\r\nyellow;|0 -203 1 28 26 178\r\nred;|0 -203 1 28 26 179\r\nblue;|0 -203 1 28 26 180',1,407,33129,0,00000000000000,NULL);
INSERT INTO `RemoteMapping` VALUES (7,'Non Pluto remote','r',NULL,'Down;|0 -106 1 201\r\nUp;|0 -106 1 200\r\nLeft;|0 -106 1 202\r\nRight;|0 -106 1 203\r\nok;|0 -106 1 190',7,289,33129,0,00000000000000,NULL);
INSERT INTO `RemoteMapping` VALUES (8,'Computing full screen','C',NULL,'Power;|0 -201 1 69 50 generic_app & 0 -203 1 5 3 <%=M%>\r\nDown;|0 -203 1 28 26 2\r\nUp;|0 -203 1 28 26 1\r\nLeft;|0 -203 1 28 26 3\r\nRight;|0 -203 1 28 26 4\r\nok;|0 -203 1 28 26 5\r\n',8,289,33129,0,00000000000000,NULL);
INSERT INTO `RemoteMapping` VALUES (9,'Navigable OSD on m/d','N',NULL,'Down;|0 -106 1 201\r\nUp;|0 -106 1 200\r\nLeft;|0 -106 1 202\r\nRight;|0 -106 1 203\r\nok;|0 -106 1 190\r\nrecorded_tv;RecTV;|0 -106 1 761',9,408,33129,0,00000000000000,NULL);
INSERT INTO `RemoteMapping` VALUES (10,'Computing list','c',NULL,'Power;|0 -201 1 69 50 generic_app',10,289,33129,0,00000000000000,NULL);
INSERT INTO `RemoteMapping` VALUES (11,'Media app full screen','f',NULL,'Down;|0 -203 1 28 26 5\r\nUp;|0 -203 1 28 26 5\r\nLeft;|0 -203 1 28 26 5\r\nRight;|0 -203 1 28 26 5\r\nok;|0 -203 1 28 26 5\r\nrecorded_tv;RecTV;|0 -106 1 761\r\n',11,408,33129,0,00000000000000,NULL);
INSERT INTO `RemoteMapping` VALUES (12,'B&O Navigable OSD','N','B','Ch+;SkipForward;chup;skipfwd;ChanUp;SkipNext;|0 -106 1 200\r\nCh-;Replay/SkipBackward;chdown;skipback;ChanDown;SkipPrior;|0 -106 1 201\r\nfastforward;Forward;|0 -106 1 203\r\nRewind;|0 -106 1 202\r\nPause;|0 -106 1 190\r\n',12,289,33129,0,00000000000000,NULL);