I think the meaning of “捨てがたく” would be clear to you if it is written with kanji and kana as “捨て難く.” It can be translated as “hard to leave behind”, “difficult to throw away”, or “hard to forego”, and so forth as can be seen from example sentences below, which I picked up from 英辞郎 and Japanese translations of English novels to let you understand the nuance of 捨てがたい.”

1. 確かに友情や愛情などの「形而上」のものも切り捨て難い。しかし、私の毎日の生活に密接にかかわるスーパーマーケットと本屋は現実的にもっと切り捨て難い。
Of course, the more metaphysical things like friendship and love would also be hard to leave behind, but supermarkets and bookstores are so closely tied in with our everyday lives that in reality they are more difficult to throw away.
【出典】英辞郎；Hiragana Times, 1991年10月号（株式会社ヤック企画）

2. 盆栽の素材の代表格は松であるが、常緑樹、落葉樹、花や実のなる樹など多くのものを盆栽に仕上げることができ、それぞれに捨て難い味わいがある。
Pine trees are typical bonsai subjects; but many other types of trees can be used, such as various evergreens and deciduous trees, and trees that blossom and bear fruit, and each has its own unique attractiveness.
【出典】英辞郎

3. -- evidently finding himself unable to forego altogether the use of the pen, the veteran had set down, as humour bade him, a thought, a reminiscence, a bit of reverie, a description of his state of mind, and so on, --
確かに、筆を執るわざを全く捨てがたいものに感じて、この老作家は、気のむくままに、感想や、追憶や、瞑想の一端や、心境の一節などを書きつけて、――
【出典】 Private Papers of Henry Ryecroft; translated by中西信太郎)

4. But spruce groves are seductive and yellow nuts of gum beguiling; they picked and loitered and strayed;
しかし、森には人をひきつけるものがあったし、黄色いやにの塊もすてがたいものがあったので、生徒達はつみとったり、ぶらぶらしたり、あてもなくさまよったりした。
【出典】Anne of Green Gables; translated by 神山妙子）